diff --git "a/D9E1T4oBgHgl3EQfEQOb/content/tmp_files/load_file.txt" "b/D9E1T4oBgHgl3EQfEQOb/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/D9E1T4oBgHgl3EQfEQOb/content/tmp_files/load_file.txt" @@ -0,0 +1,1600 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf,len=1599 +page_content='A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We strengthen the classical approximation theorems of Weierstrass, Runge and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In particular, when approximating a function f on a compact set K, all the critical points of our approximants lie close to K, and all the critical values lie close to f(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Our proofs rely on extensions of (1) the quasiconformal folding method of the first author, and (2) a theorem of Carath´eodory on approximation of bounded analytic functions by finite Blaschke products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Introduction The following is Runge’s classical theorem on polynomial approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' [Run85] Let f be a function analytic on a neighborhood of a compact set K ⊂ C, and suppose C \\ K is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For all ε > 0, there exists a polynomial p so that ||f − p||K := sup z∈K |f(z) − p(z)| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This famous result does not say much about what the polynomial approximant p looks like off the compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For various applications, it would be useful to understand the global behavior of p and, in particular, the location of the critical points and values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' To this end, we state our first result (Theorem A below) after introducing the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Notation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For any compact set K ⊂ C we denote by fill(K) the union of K with all bounded components of C \\ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We say K is full if C \\ K is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We let CP(f) denote the set of critical points of an analytic function f, and let CV(f) := f(CP(f)) denote its critical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A domain in �C is an open, connected subset of �C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (Polynomial Runge+) Let K ⊂ C be compact and full, D a domain containing K, and suppose f is a function analytic in a neighborhood of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then for all ε > 0, there exists a polynomial p so that ||p − f||K < ε and: (1) CP(p) ⊂ D, (2) CV(f|K) ⊂ CV(p) ⊂ fill{z : d(z, f(K)) ≤ ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Primary: 30C10, 30C62, 30E10, Secondary: 41A20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' uniform approximation, polynomials, rational functions, Blaschke products, Runge’s Theorem, Weierstrass’s Theorem, Mergelyan’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The first author is partially supported by NSF Grant DMS 1906259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='02888v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='CV] 7 Jan 2023 2 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Analogous improvements of the polynomial approximation theorems of Mergelyan and Weierstrass will be stated and proved in Section 9 (see Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8 and Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' When K is not full, uniform approximation by polynomials is not always possible, and so we turn to rational approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We denote the Hausdorff distance between two sets X, Y , by dH(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (Rational Runge+) Let K ⊂ C be compact, D a domain containing K, f a function analytic in a neighborhood of K, and suppose P ⊂ �C \\ K contains exactly one point from each component of �C \\ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then there exists P ′ ⊂ P so that for all ε > 0, there is a rational function r so that ||r − f||K < ε and: (1) dH(r−1(∞), P ′) < ε and |r−1(∞)| = |P ′|, (2) CP(r) ⊂ D, (3) CV(f|K) ⊂ CV(r) ⊂ fill{z : d(z, f(K)) ≤ ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The behavior of p off K is of particular interest in applications, such as in complex dy- namics where approximation results have been used to prove the existence of various dy- namical behaviors for entire functions (see, for example, [EL87], [BT21], [MRW21], [ERS22], [MRW22], [BEF+22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' However, not understanding the critical points and values of p means it has not been known whether these behaviors can occur within restricted classes of en- tire functions, such as the well studied Speiser or Eremenko-Lyubich classes (see the survey [Six18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Our approach is based on two ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The first is to show that on a compact subset K ⊂ C of a finitely connected domain Ω ⊂ C, any bounded analytic function can be approximated uniformly by an analytic function B : Ω → C having the property that |B| is constant on each component of ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This extends a classical theorem of Carath´eodory [Car54] concerning finite Blaschke products on the unit disk to more general regions, and may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The second idea is an extension of quasiconformal folding, a type of quasiconformal surgery introduced in [Bis15], to extend the (generalized) Blaschke product B from Ω to a quasireg- ular mapping g : �C → �C with specified poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The map g may be taken close to holomorphic in a suitable sense, and so the Measurable Riemann Mapping Theorem (MRMT for brevity) will imply there is a quasiconformal mapping φ so that g ◦ φ−1 is the desired polynomial or rational approximant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This approach yields not only information on the critical points and values of the approx- imants as in Theorems A and B, but more broadly a detailed description of the geometric structure of these approximants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We end the introduction by describing this geometric structure in a few cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' First we introduce some more notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Notation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let D ⊂ �C be a simply connected domain so that ∞ ̸∈ ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We let ψD : E → D denote a Riemann mapping, where E = D is D is bounded and E = D∗ := �C\\D if D is unbounded, in which case we specify ψD(∞) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 3 First consider the case when K is full and connected, and f is holomorphic in a neighbor- hood of K satisfying ||f||K < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let Ω, Ω′ be analytic Jordan domains containing K, f(K), respectively, so that f is holomorphic in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then the mapping F := ψ−1 Ω′ ◦ f ◦ ψΩ : D → D is holomorphic, and by Carath´eodory’s theorem for the disk (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1), there is a finite Blaschke product b : D → D that approximates F on the compact set ψ−1 Ω (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Therefore B := ψΩ′ ◦ b ◦ ψ−1 Ω : Ω → Ω′ is a holomorphic function that approximates f on K, and moreover B restricts to an analytic, finite-to-1 map of Γ := ∂Ω onto Γ′ := ∂Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In this paper, we will show that B can be approximated on Ω by a polynomial p so that p−1(Γ′) is an approximation of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' More precisely, p−1(Γ′) is connected, and consists of a finite union of Jordan curves {γj}n 0 bounding pairwise disjoint Jordan domains {Ωj}n 0 (see Figure 1): the {Ωj}n 0 are precisely the connected components of p−1(Ω′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' There is one “large” component Ω0 that approximates Ω in the Hausdorff metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The other components {Ωj}n 1 can be made as small as we wish and to lie in any given neighborhood of ∂K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, the collection {Ωj}n 0 forms a tree structure with any two boundaries ∂Ωj, ∂Ωk either disjoint or intersecting at a single point, and with Ω0 as the “root” of the tree as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let Ω∞ denote the unbounded component of C \\ p−1(Γ′), so that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) C \\ p−1(Γ′) = Ω0 ⊔ � ⊔n j=1Ωj � ⊔ Ω∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recalling Notation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3, the polynomial p has the following simple structure with respect to the domains in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (1) p(Ω0) = Ω′ and ψ−1 Ω′ ◦ p ◦ ψΩ0 is a finite Blaschke product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (2) p(Ωj) = Ω′ and p is conformal on Ωj for 1 ≤ j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (3) p(Ω∞) = C \\ Ω′ and p = ψC\\Ω′ ◦ (z �→ zm) ◦ ψ−1 Ω∞ on Ω∞ for m = deg(p|Ω0) + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In other words, up to conformal changes of coordinates, p is simply a Blaschke product in Ω0, a conformal map in each Ωj, 1 ≤ j ≤ n, and a power map z �→ zm in Ω∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The only finite critical points of p are either in Ω0, or at a point where two of the curves (γj)n j=1 intersect, in which case the corresponding critical value lies on ∂Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Next suppose K is connected, but C \\ K has more than one component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In this case, in order to prove Theorem B, we will need to let Ω be a multiply connected analytic domain containing K, and Ω′ an analytic Jordan domain containing f(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We cannot proceed as in the case C\\K is connected, however, without a multiply connected version of Carath´eodory’s Theorem for the disk (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The usual proof of Carath´eodory’s Theorem is based on power series, and it does not extend to the multiply connected setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' However, there is an alternate proof based on potential theory that does extend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We now briefly sketch the idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 4 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK K p Ω∞ f(K) Γ′ Ω0 {Ωj}n 1 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This figure illustrates the geometry of the approximant p in Theorem A when K is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The notation is explained in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Both domain and co- domain are colored so that regions with the same color correspond to one another under p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose f is holomorphic in a simply connected domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' To simplify matters, suppose f(D) is compactly contained in D \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then u := − log |f| is a positive harmonic function on D, and u is also bounded and bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus u is the Poisson integral of its boundary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The Poisson kernel Px(z) associated to any point x ∈ ∂D is a limit of normalized Green’s functions on D: Px(z) ≈ G(z, wn)/G(0, wn) with wn → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Approximating the Poisson integral by a Riemann sum gives an approximation of u on any compact set K ⊂ D by a sum of Green’s functions H(z) := � j G(z, wj) with poles distributed on {z : d(z, ∂D) = δ}, and with δ as small as we wish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Figure 2 shows the function u(x, y) = 1 2 + xy being approximated by a sum of 25 Green’s functions on D = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since D is simply connected, H has a well defined harmonic conjugate �H, and after adding a constant to �H if necessary, B := exp(−H − i �H) approximates f on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, B satisfies: (1) B is holomorphic on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (2) |B| extends continuously to a constant function on ∂D, with ||B||∂D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We call such a function B on a (not necessarily simply connected) domain D a (generalized) finite Blaschke product on D (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' When D = D this definition coincides with A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' On the left is the positive harmonic function u(x, y) = 1 2 + xy on the unit disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' On the right is a sum of 25 Green’s functions with poles on the circle of radius .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' On D(0, 1 2) the two functions agree to within .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' As expected, the poles are closer together where u is large and farther apart where u is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' the usual definition of Blaschke product, and the above argument yields Carath´eodory’s The- orem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In fact, the above argument yields several technical improvements of Carath´eodory’s Theorem (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) which we will need in order to prove Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This argument generalizes to finitely connected domains D, except that the sum of Green’s functions H may not have a well defined harmonic conjugate (even modulo 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will fix this by adding a small harmonic function h that is constant on each boundary component of D and whose periods match the periods of −H around each boundary component of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Exponentiating the sum of the modified function H + h with its harmonic conjugate (now well-defined modulo 2π) then gives a (generalized) finite Blaschke product B which approximates the given function f on the desired compact set K ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This gives a version of Carath´eodory’s theorem on finitely connected domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By choosing H correctly, we can take ∥h∥D as small as we wish, and hence |B| is constant on each connected component of ∂D, with values that can all be taken as close to 1 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Now we return to the description of our rational approximant in the case that K is connected, but C\\K has more than one component, recalling that Ω is a multiply connected analytic domain containing K, and Ω′ is an analytic Jordan domain containing f(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By the multiply connected version of Caratheodory’s Theorem, there exists a (generalized) finite Blaschke product b approximating ψ−1 Ω′ ◦ f on K, so that B := ψΩ′ ◦ b is a holomorphic map approximating f on K, and B restricts to an analytic, finite-to-1 mapping of each component of Γ := ∂Ω onto ψΩ′(|z| = t) for t = 1 or t ≈ 1, where t may depend on the component of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 ~ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='56 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK We will show B can be approximated on Ω by a rational map r so that each component of ∂Ω can be approximated by a component of r−1 ◦ ψΩ′(|z| = t) for t as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' These components of r−1 ◦ ψΩ′(|z| = t) bound Jordan domains which form a decomposition of the plane as in the previously described polynomial setting, and in the interior of each such domain again r behaves either as a (generalized) finite Blaschke product, a conformal mapping, or a power mapping (up to conformal changes of coordinates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lastly, the case when K has more than one connected component is more intricate, and we will leave the precise description to later in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (Briefly, quasiconformal folding is applied not just along the boundary of a neighborhood of K, but also along specially chosen curves that connect different connected components of this neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=') We remark that while Theorem A strictly improves on Runge’s Theorem on polynomial approximation, the relationship between Theorem B and Runge’s Theorem on rational ap- proximation is more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Both show existence of rational approximants, and only The- orem B describes the critical point structure of the approximant, however the poles of the approximant in Theorem B are specified only up to a small perturbation, whereas in Runge’s Theorem they are specified exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We do not know whether it is necessary to consider per- turbations of P ′ in Theorem B, or if the improvement r−1(∞) = P ′ is possible (a related problem appears in [BL19], [DKM20], [BLU], where it is known no such improvement is possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The authors would like to thank Malik Younsi and Oleg Ivrii for useful discussions related to this manuscript, and Xavier Jarque for his comments on a preliminary version of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Interior Approximation The following classical result of Carath´eodory (referenced in the introduction) allows for approximation by Blaschke products in simply-connected domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' ([Car54]) Let f : D → C be holomorphic and suppose ||f||D ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then there exists a sequence of finite Blaschke products on D converging to f uniformly on compact subsets of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 is elementary and may be found, for example, in Theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 of [Gar81] or Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 of the survey [GMR17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In order to prove Theorem A, we will need to prove a version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 in which the Blaschke products satisfy certain boundary regularity conditions, and in order to prove Theorem B, we will need to prove a multiply- connected version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' These improvements are stated below in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' First we need several definitions: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let D ⊂ C be a finitely connected domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We say a non-constant holo- morphic function B : D → C is a finite Blaschke product on D if |B| extends continuously to a non-zero, constant function on each component of ∂D and ||B||D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 7 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' When D = D, the definition above corresponds with the usual definition of finite Blaschke product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For a finite Blaschke product B on a finitely connected domain D, we let IB denote the connected components of ∂D \\ {z : B(z) ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In other words, IB are the preimages (under B) of the open upper and lower half-circle components of B(∂D) ∩ H, B(∂D) ∩ (−H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will frequently be dealing with sequences of finite Blaschke products (Bn)∞ n=1 on D, in which case we abbreviate IBn by In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We call a domain D ⊂ C an analytic domain if D is finitely connected, and each component of ∂D is an analytic Jordan curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We remark that a boundary component of an analytic domain D cannot be a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let D ⊂ C be an analytic domain, suppose K ⊂ D is compact, and let f be a function analytic in a neighborhood of D satisfying ||f||D < 1 and {z : f(z) = 0}∩∂D = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then there exists M < ∞ and a sequence (Bn)∞ n=1 of finite Blaschke products on D satisfying: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) inf z∈∂D |Bn(z)| n→∞ −−−→ 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) ||Bn − f||K n→∞ −−−→ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) sup I∈In diam(I) n→∞ −−−→ 0, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) sup I,J∈In diam(I)/diam(J) < M and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) sup I∈In supz∈I |B′ n(z)| infz∈I |B′ n(z)| < M for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 will suffice for the proofs of Theorems A and B, although we prove a slightly stronger result in Section 12 (see Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 will be delayed until Sections 10-12, which are independent of Sections 2-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We now turn to applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 to produce Blaschke approximants as described in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Given a Jordan curve γ ⊂ C, we denote the bounded component of �C \\ γ by int(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We refer to Figure 3 for a summary of the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For the remainder of this section, we will fix a compact set K, an analytic domain D containing K, and a function f holomorphic in a neighborhood of D satisfying ||f||D < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Fix ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We assume that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) d(z, K) < ε/2 and d(f(z), f(K)) < ε/2 for every z ∈ ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We let γ be an analytic Jordan curve surrounding f(D) such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) dist(w, f(K)) < ε for every w ∈ γ, and let Ψ : D → int(γ) denote a Riemann mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 8 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This figure illustrates Notations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The vertices pictured on ∂D are B−1(±1), and the components IB of Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4 are the edges along ∂D connecting these vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By enlarging D slightly we may ensure {z : Ψ−1 ◦ f(z) = 0} ∩ ∂D = ∅, so that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 applies to the triple D, K, Ψ−1 ◦ f to produce M < ∞ and a sequence of finite Blaschke products (Bn)∞ n=1 on D satisfying the conclusions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Note that in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9 we are applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 to Ψ−1 ◦ f (and not f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This will ensure that the critical values of the approximant Ψ ◦ Bn ≈ f are close to fill(f(K)) as needed for Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will now quasiconformally perturb the sequence (Bn) so as to ensure that we can later prove the conclusion CV(f|K) ⊂ CV(r) of Theorem B: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We may take the sequence (Bn)∞ n=1 of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9 so that it also satisfies: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8) CV(Ψ−1 ◦ f|K) ⊂ CV(Bn) for all large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let K′ ⊂ D be a compact set satisfying K ⊂ int(K′) ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 to the triple D, K′, Ψ−1 ◦ f to obtain a sequence of Blaschke products (Bn)∞ n=1 on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let z ∈ CP(f|K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9) ||Bn − Ψ−1 ◦ f||K′ n→∞ −−−→ 0, by Hurwitz’s Theorem there exists a sequence (wn z )∞ n=1 such that B′ n(wn z ) = 0 and wn z n→∞ −−−→ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let r < 1 be so that ψ−1 ◦ f(K) ⊂ rD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Define a homeomorphism hn : D → D to be an interpolation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10) hn(Bn(wn z )) := Ψ−1 ◦ f(z) for all z ∈ CP(f|K), and int() f(K) Y Izl<1 D K B ~ ofA GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 9 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11) hn(z) = z for r ≤ |z| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9), we have that |Bn(wn z ) − Ψ−1 ◦ f(z)| n→∞ −−−→ 0 for all z ∈ CP(f|K), and hence hn may be taken to satisfy: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12) ||(hn)z/(hn)z||D n→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By the Measurable Riemann Mapping Theorem (see [Ahl06]), there is a quasiconformal φn : D → D so that: Bn := hn ◦ Bn ◦ φ−1 n is holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We normalize each φn by specifying φn(p) = p and φ′ n(p) > 0 for some r < |p| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Note that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13) Bn(∂D) ⊂ {z : r ≤ |z| ≤ 1} for large n by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1), so it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11) that Bn is a Blaschke product on D for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We claim the sequence (Bn) satisfies the conclusions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6, as well as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Indeed, we have φn(wn z ) ∈ CP(Bn) and Bn(φn(wn z )) = Ψ−1 ◦ f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12), we have that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='14) ||φn(z) − z||D n→∞ −−−→ 0, and hence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' To prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5), we first note that since (hn)z = 0 in r < |z| < 1, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13) that φn extends to a holomorphic function in a neighborhood U of ∂D, where U does not depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='14) and the Cauchy integral formula, it follows that φ′ n(z) n→∞ −−−→ 1 uniformly for z ∈ ∂D and hence we deduce (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Similarly, φ′′ n(z) n→∞ −−−→ 0 uniformly for z ∈ ∂D and so (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Recall from the introduction that we plan to extend the definition of the approximant Ψ ◦ Bn ≈ f from D to all of C, where we recall Ψ was defined in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' To this end, it will be useful to define the following graph structure on ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For any n ∈ N, we define a set of vertices on ∂D by Vn := (Bn|∂D)−1(R), where each vertex v is labeled black or white according to whether Bn(v) > 0 or Bn(v) < 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The curve ∂D will be considered as a graph with edges defined by In (recall from Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4 that In is precisely the collection of components of ∂D \\ Vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will sometimes write Dn in place of D when we wish to emphasize the dependence of the graph ∂D on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We define a holomorphic mapping gn in D by the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='15) gn(z) := Ψ ◦ Bn(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 10 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK In Sections 3-7 we will quasiregularly extend the definition of gn to C, and then in Section 9 we apply the MRMT to produce the rational approximant of Theorem B as described in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall that in Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7, we fixed ε > 0, a compact set K contained in an analytic domain D, and a function f holomorphic in D (we note ε, K, D, f also satisfied extra conditions specified in Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The objects γ, Ψ, Bn, Vn, gn we then defined in this section were determined by our initial choice of ε, K, D, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In future sections, it will be useful to think of γ, Ψ, Bn, Vn, gn as defining functions which take as input some quadruple (ε, K, D, f) (for any ε, K, D, f as in Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7), and output whatever object we defined in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For instance, Vn defines a function which takes as input any (ε, K, D, f) as in Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7 and outputs (via Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11) a set of vertices Vn(ε, K, D, f) on ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Similarly, Bn takes as input any (ε, K, D, f) as in Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7 and outputs (via Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10) a Blaschke product Bn(ε, K, D, f) on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Likewise for γ, Ψ, gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Quasiconformal Folding Given a compact set K ⊂ C and a function f holomorphic in a domain D containing K, we showed in Section 2 how to approximate f by a holomorphic function gn defined in D (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover gn is just a Blaschke product in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If f is a function holomorphic in an arbitrary analytic neighborhood U (where U need not be connected) of a compact set K, then one can apply the results of Section 2 to each component of U which intersects K (this is done precisely in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1): this yields a holomorphic approximant of f defined in a finite union of domains, so that the approximant is just a finite Blaschke product on each domain (recall Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In Sections 3-7, we will build the apparatus necessary to systematically extend this holomorphic approximant to a quasiregular function of C which is holomorphic outside a small set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It was convenient to assume in Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7 that the compact set K was covered by a single domain D, however we now begin to work more generally: Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We refer to Figure 6 for a summary of the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Throughout Sections 3-7, we will fix ε > 0, a compact set K ⊂ C, a domain D containing K, a disjoint collection of analytic domains (Di)k i=1 such that K ⊂ U := ∪iDi ⊂ D, and a function f holomorphic in a neighborhood of U satisfying ||f||U < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We assume that the following analog of Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) holds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) d(z, K ∩ Di) < ε/2 and d(f(z), f(K ∩ Di)) < ε/2 for all z ∈ ∂Di and 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Applying the methods of the previous section to each component Di of U, we can define a sequence of finite Blaschke products (Bn)∞ n=1 on each Di (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will let Bn denote the corresponding function defined on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In particular, (Bn)∞ n=1 gives the following definition of vertices on the boundary of U := ∪iDi (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 11 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Illustrated is the Definition of the domain V in the proof of Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For every n ∈ N, we define a set of vertices Vn on ∂U by Vn := k� i=1 Vn(ε, K ∩ Di, Di, f|Di) = k� i=1 (Bn|∂Di)−1(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We now extend the graph structure on ∂U by connecting the different components of U by curves {Γi}k−1 i=1 in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 below, and defining vertices along these curves in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will need to prove a certain level of regularity for these curves and vertices in order to ensure that the dilatations of quasiconformal adjustments we will make later do not degenerate as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will denote the curves by {Γi}k−1 i=1 , and we remark that the curves depend on n, although we suppress this from the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For each n ∈ N, there exists a collection of disjoint, closed, analytic Jordan arcs {Γi}k−1 i=1 in (�C \\ U) ∩ D satisfying the following properties: (1) Each endpoint of Γi is a vertex in Vn, (2) Each Γi meets ∂U at right angles, (3) U ∪ (∪iΓi) is connected, and (4) For each 1 ≤ i ≤ k − 1, the sequence (in n) of curves Γi has an analytic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The set (�C \\ U) ∩ D must contain at least one simply-connected region V with the property that there are distinct i, j with both ∂V ∩∂Di and ∂V ∩∂Dj containing non-trivial arcs (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3), for all sufficiently large n both ∂V ∩ ∂Di, ∂V ∩ ∂Dj contain vertices of Vn which we denote by vi ∈ ∂Di, vj ∈ ∂Dj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Consider a conformal map φ : D → V , and define Γ1 to be the image under φ of the hyperbolic geodesic connecting φ−1(vi), φ−1 i (vj) in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We now proceed recursively, making sure at step l we pick a V which connects two com- ponents of U not already connected by a Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', Γl−1, and so that V is disjoint from Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', Γl−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The curves Γi satisfy conclusions (1)-(3) of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We may ensure that for each 1 ≤ i ≤ k − 1, the sequence (in n) of curves Γi has an analytic limit by choosing vi, vj above to converge as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ D D2 V IJ D112 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Illustrated is Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Consider the vertices Vn ⊂ ∂U of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will augment Vn to include vertices on the curves (Γi)k−1 i=1 as follows (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let Γ ∈ (Γi)k−1 i=1 denote both the curve as a subset of C and the arclength parameterization of the curve, and suppose Γ connects vertices Γ(0) = vi ∈ ∂Di, Γ(length(Γ)) = vj ∈ ∂Dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For k = i, j, let εk denote the minimum length of the two edges with endpoint vk in ∂Dk, and suppose without loss of generality εj < εi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let l be so that εj/2 ≤ εi/2l ≤ 2εj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We place vertices at Γ(εi/2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', Γ(εi/2l), and we place vertices along Γ([εi/2l, length(Γ)]) at equidistributed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We can label the vertices black/white along Γ so that vertices connect only to vertices of the opposite color by adding one extra vertex at the midpoint of the segment having vj as an endpoint, if need be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We introduce the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Throughout Sections 3-7, we will let Ω denote a fixed (arbitrary) component of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) �C \\ � U ∪ k−1 � i=1 Γi � , and p ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Note that Ω is simply connected by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Denote D∗ := �C\\D, and let σ denote any conformal mapping (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) σ : D∗ → Ω satisfying σ(∞) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For z ∈ Ω, we define τ(z) := σ−1(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The map τ induces a partition of T which we denote by Vn := τ(Vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' D1 E1 82A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 13 |z|>1 Ω τ 3 D D2 Γ2 1Γ Γ3 D4 D1 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This figure illustrates Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 and Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' As pictured, U has four components (Di)4 i=1 which are connected by curves (Γi)3 i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall K ⊂ U (the compact set K is not shown in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The unbounded component Ω of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) is pictured in dark grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The map τ : Ω → D∗ is a conformal mapping, and sends the vertices on ∂Ω to (possibly unevenly spaced) vertices on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will sometimes write Ωn, D∗ n in place of Ω, D∗, respectively, when we wish to emphasize the dependence of the vertices Vn ⊂ ∂Ω, Vn ⊂ ∂D∗ on the parameter n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For the graph ∂Ωn, we have: max{diam(e) : e is an edge of ∂Ωn} n→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) and Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ As explained in the introduction, in order to prove uniform approximation in Theorem B, we will need to prove that our quasiregular extension is holomorphic outside a region of small area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This will usually mean proving the following condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose V ⊂ C is an analytic domain, and ∂V is a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We say a quasiregular mapping φ : V → φ(V ) is C-vertex-supported if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) supp(φz) ⊂ � e∈∂V {z : dist(z, e) < C · diam(e)} (see Figure 7), where the union in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) is taken over all edges e on ∂V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It will also be useful to have the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose e, f are rectifiable Jordan arcs, and h : e → f is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We say that h is length-multiplying on e if the push-forward (under h) of arc-length measure on e coincides with the arc-length measure on f multiplied by length(f)/length(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 14 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Shown as a black curve is part of a graph G, and in light gray the neighborhood ∪e∈G{z : dist(z, e) < C · diam(e)} of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' First we will adjust the conformal map τ so as to be length-multiplying along edges of ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall the vertices Vn ⊂ T defined in Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For every n, there is a K-quasiconformal mapping λ : D∗ n → D∗ n so that: (1) λ is C-vertex-supported for some C > 0, (2) λ(z) = z on Vn and off of supp(λz), (3) λ ◦ τ is length-multiplying on every component of ∂Ω \\ Vn, (4) C, K do not depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This is a consequence of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 of [Bis15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Indeed, recall τ := σ−1 and consider the 2πi-periodic covering map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) φ := σ ◦ exp : Hr �→ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The map φ induces a periodic partition φ−1(Vn) of ∂Hr which has bounded geometry (see the introduction of [Bis15], or Section 2 of [BL19]) with constants independent of n by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3(2) and Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 of [Bis15] applies to produce a 2πi-periodic, C vertex-supported, and K-quasiconformal map β : Hr → Hr so that φ ◦ β is length-multiplying on edges of Hr, and C, K are independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, the inverse β−1 ◦ log ◦τ is length-multiplying, and since exp is length-multiplying on vertical edges, the well-defined map λ := exp ◦β−1 ◦ log : D∗ → D∗ satisfies the conclusions of the Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ The main idea in defining the quasiregular extension in Ω is to send each edge of ∂Di to the upper or lower half of the unit circle by following λ ◦ τ with a power map z �→ zn A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 15 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This figure illustrates the Folding Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13 and Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The simply connected domain Ω′ n is obtained by removing from Ω certain trees based at the vertices along ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' of appropriate degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The main difficulty in this approach, however, is that the images of different edges of ∂Di under λ ◦ τ may differ significantly in size, so that there is no single n with z �→ zn achieving the desired behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The solution is to modify the domain Ω by removing certain “decorations” from the domain Ω, so that each edge of ∂Di is sent to an arc of roughly the same size under λ ◦ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This is formalized below in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13 (see also Figures 8, 9), and is an application of the main technical result of [Bis15] (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The “decorations” are the trees in the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let V ⊂ T be a discrete set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We call a domain W ⊂ D∗ a tree domain rooted at V if W consists of the complement in D∗ of a collection of disjoint trees, one rooted at each vertex of V (see the center of Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For m ∈ N, we let Z± m := {z ∈ T : zm = ±1}, Zm := Z+ m ∪ Z− m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In other words, Z+ m denotes the mth roots of unity, and Z− m the mth roots of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For every n, there exists a tree domain Wn rooted at Vn, an integer m = m(n), and a K-quasiconformal mapping ψ : Wn → D∗ so that: (1) ψ is C-vertex-supported for some C > 0, and ψ(z) = z off of supp(ψz), (2) on any edge e of ∂Wn ∩ T, ψ is length-multiplying and ψ(e) is an edge in T \\ Zm, (3) for any edge e of ∂Wn ∩D∗, ψ(e) consists of two edges in T\\Zm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, if x ∈ e, the two limits limWn∋z→x ψ(z) ∈ T are equidistant from Z+ m, and from Z− m, and (4) C, K do not depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We consider the 2π-periodic covering map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) φ := σ ◦ λ ◦ exp ◦(z �→ −iz) : H �→ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Izl>1 ov16 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For any x ∈ ∂Wn ∩ D∗, there are two limits limWn∋z→x ψ(z) ∈ T as illustrated in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13(3) says that these two limits are equidistant from the nearest black vertex, and are equidistant from the nearest white vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' inducing a periodic partition φ−1(Vn) of ∂H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4), Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4, and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10(2), any two edges of H have comparable lengths with constant independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' There- fore, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 of [Bis15] applies to yield a 2π-periodic K-quasiconformal map Ψn of H onto a subdomain Ψn(H) ⊊ H, with K independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We let Wn := exp(−iΨn(H)) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) ψ := exp ◦ − iΨ−1 n ◦ i log : Wn → D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) is well-defined, and the conclusions of the theorem follow from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 of [Bis15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will use the notation Ω′ n := (λ ◦ τ)−1(Wn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Annular Interpolation Between the Identity and a Conformal Mapping Recall from Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 that we have fixed ε > 0, a compact set K, disjoint analytic domains (Di)k i=1 so that U := ∪iDi contains K, and f holomorphic in a neighborhood of U with ||f||U < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In this section, we briefly define two useful interpolations in Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since the domain Di contains the compact set K∩Di, the definitions and results of Section 2 apply to (ε, K ∩ Di, Di, f|Di) for each 1 ≤ i ≤ k (see Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13 applies to define (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We define the Jordan curve (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) γi := γ(ε, K ∩ Di, Di, f|Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recalling that int(γi) denotes the bounded component of �C \\ γi, we define (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) Ψi := Ψ(ε, K ∩ Di, Di, f|Di) to be a Riemann mapping Ψi : D → int(γi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lastly, we define the finite Blaschke products (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) Bn := Bn(ε, K ∩ Di, Di, f|Di) on Di, A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 17 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Illustrated is the map ηΨ i : D∗ → �C \\ Ψi(riD) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 in the case ri = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The dotted circle on the right depicts the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' where we suppress the dependence of (Bn)∞ n=1 on i from the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall that in Section 3, we defined curves {Γi}k−1 i=1 connecting the domains Di, and in Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 we fixed a component Ω of the complement of U ∪ ∪k−1 i=1 Γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' After relabeling the (Di)k i=1 if necessary, there exists 1 ≤ ℓ ≤ k so that ∂Di ∩ ∂Ω ̸= ∅ if and only if i ≤ ℓ (see Figure 6 for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For each 1 ≤ i ≤ ℓ, note that the intersection ∂Di ∩ ∂Ω consists of a single Jordan curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We let ri := |Bn(∂Di ∩ ∂Ω)|, so that 1 − ε ≤ ri ≤ 1 by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The two interpolations we will need are given in Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3, we define an interpolation ηΨ i between z �→ z on |z| = 2 with z �→ Ψi(riz) on |z| = 1 (see Figure 10), and in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4 we modify ηΨ i to define a map ηi so that ηi(z) = ηi(z) for |z| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For each 1 ≤ i ≤ ℓ, there is a quasiconformal mapping ηΨ i : D∗ → �C \\ Ψi(riD) satisfying the relations: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) ηΨ i (z) = z for |z| ≥ 2 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) ηΨ i (z) = Ψi(riz) for all |z| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, if Di, Dj for 1 ≤ i, j ≤ ℓ are connected by one of the curves (Γi)k−1 i=1 , then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) ηΨ i ([−2, −1]) ∩ ηΨ j ([1, 2]) = ηΨ i ([1, 2]) ∩ ηΨ j ([−2, −1]) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The existence of ηΨ i satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) follows from a standard lemma on the extension of quasisymmetric maps between boundaries of quasiannuli (see, for instance, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='30(b) of [BF14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) fails for the collection (ηΨ i )i∈I thus defined, we can renormalize the conformal mappings (Ψi)ℓ i=1 appropriately (to rotate the points Ψi(±ri) along the curve Ψi(riT)), and post-compose a subcollection of the ηΨ i by diffeomorphisms of DV n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='18 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK 2D \\ Ψi(riD) so that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) is satisfied for the composition when i, j ∈ I, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) still hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For each 1 ≤ i ≤ ℓ, there is a quasiconformal mapping ηi : D∗ → C \\ Ψi([−ri, ri]) satisfying the relations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) ηi(z) = z for |z| ≥ 2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8) ηi(z) = ηi(z) for |z| = 1, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9) ηi(z) = ηΨ i (z) for z ∈ R ∩ D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Define γ+ i := ηΨ i (∂(A(1, 2) ∩ H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let η be a quasisymmetric mapping of T∩H onto [−1, 1] fixing ±1 (one can take η := M|T∩H where M is a Mobius transformation mapping −1, 1, i to −1, 1, 0, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Define a mapping g on γ+ i by: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10) g(z) := � Ψi ◦ η ◦ Ψ−1 i (z) z ∈ Ψi(T ∩ H) z otherwise Since g is a quasisymmetric mapping, a standard lemma on extension of quasisymmetric maps between boundaries of quasidisks (see, for instance, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='30(a) of [BF14]) implies that g may be extended to a quasiconformal mapping of ηΨ i (A(1, 2) ∩ H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Define g similarly in ηΨ i (A(1, 2) ∩ (−H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We let ηi := g ◦ ηΨ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It is then straightforward to check that ηi satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4 define 2ℓ many quasiconformal mappings: {ηΨ i }ℓ i=1 and {ηi}ℓ i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The definition of the mappings ηΨ i , ηi depend on the objects ε, K, (Di)k i=1, f as fixed in Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1, but not on the parameter n in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus we record the trivial but important observation that the mappings {ηΨ i }ℓ i=1 and {ηi}ℓ i=1 are quasiconformal with a constant independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Annular Interpolation Between a Blaschke Product and a Power Map Recall that we have fixed ε > 0, a compact set K, disjoint analytic domains (Di)k i=1 so that U := ∪iDi contains K, and f holomorphic in a neighborhood of U with ||f||U < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The curves {Γi}k−1 i=1 connect the domains (Di)k i=1, and Ω is a component of the complement of U ∪ ∪k−1 i=1 Γi with τ : Ω → D∗ conformal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall that the domain Ω′ n was defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13 and Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='14 by removing from Ω a collection of trees rooted at the vertices along ∂Ω, and the map ψ ◦ λ ◦ τ maps Ω′ n onto D∗ (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 19 Notation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall from Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 that ∂Di ∩ ∂Ω ̸= ∅ if and only if 1 ≤ i ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Hence exactly ℓ − 1 of the curves (Γi)k−1 i=1 intersect ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By relabelling the (Γi)k−1 i=1 if necessary, we may assume Γj intersects ∂Ω if and only if 1 ≤ j ≤ ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let m = m(n) be as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' To prove our main results, we will need to modify z �→ zm in D∗ so that, roughly speaking, (z �→ zm) ◦ ψ ◦ λ ◦ τ(z) agrees with the Blaschke products Bn (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) along ∂Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This is done in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Its proof uses the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose φ1, φ2 are C1 homeomorphisms of a C1 Jordan arc e such that: (1) φ1(e) = φ2(e), (2) φ1, φ2 agree on the two endpoints of e, and (3) |φ′ 1(z)| = |φ′ 2(z)| for all z ∈ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then φ1 = φ2 on e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 is a consequence of the Fundamental Theorem of Calculus and is left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall the constant ri := |Bn(∂Di ∩ ∂Ω)| of Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For every n, there exists a locally univalent K-quasiregular mapping hn : D∗ → D∗ so that: (1) hn(z) = zm for |z| ≥ m√ 2 where m := m(n) is as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13, (2) hn ◦ ψ ◦ λ ◦ τ(z) = Bn(z)/ri for every z ∈ ∂Di and 1 ≤ i ≤ l, and (3) K is independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Fix the standard branch of log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Given an edge e ∈ ∂Di, we have by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13 that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) log ◦ψ ◦ λ ◦ τ(e) = {0} × �jπ m , (j + 1)π m � for some 0 ≤ j ≤ 2m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Denote the vertical line segment in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) by ve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let f : ve �→ e be a length-multiplying, C1 homeomorphism so that f −1 agrees with log ◦ψ ◦ λ ◦ τ on the two endpoints of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Consider the maps: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) z �→ mz for z ∈ �log 2 m � × �jπ m , (j + 1)π m � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) z �→ log ◦r−1 i Bn ◦ f for z ∈ {0} × �jπ m , (j + 1)π m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For each 1 ≤ i ��� l, the Blaschke products Bn are orientation-preserving on the unique outer boundary component of Di, and orientation-reserving on all other boundary components of Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This implies that we may choose the branch of log in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) so that the images of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) are horizontal translates of one another (recall Bn(e) is a circular arc of angle π), and the derivative of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) is strictly positive for all z ∈ ve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since the derivative of 20 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Illustrated is the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In logarithmic coordi- nates the desired interpolation is denoted φ, and hn is then defined by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) is also strictly positive, this means the linear interpolation between (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5), we have that |B′ n| is comparable at all points of e with constant independent of e and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, since f is length-multiplying and log is length-multiplying on Euclidean circles centered at 0, we conclude that the derivative of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) is comparable to m at all points of ve with constant independent of e and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, we conclude that the linear interpolation between (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) in the rectangle (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) � 0, log 2 m � × �jπ m , (j + 1)π m � is K-quasiconformal with K independent of e and n (see, for instance, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 of [MPS20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Denote the linear interpolation by φ (see Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We define (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) hn := exp ◦φ ◦ log in {z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(e)} ∩ {|z| ≤ m√ 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) defines hn(z) for z in {z : 1 ≤ |z| ≤ m√ 2} and sharing a common angle with the image under ψ ◦ λ ◦ τ of an edge on some ∂Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We finish the definition of hn by simply setting: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) hn(z) := zm in {z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(∂Ω′ n \\ (∪i∂Di))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The conclusion (1) now follows by definition of hn, and (3) follows since hn is a composition of holomorphic mappings and a K-quasiconformal interpolation where we have already noted that K is independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' h nA GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 21 We now show that conclusion (2) follows from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Fix an edge e on ∂Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall ve := log ◦ψ ◦ λ ◦ τ(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) we have that: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) hn ◦ ψ ◦ λ ◦ τ = r−1 i Bn ◦ f ◦ log ◦ψ ◦ λ ◦ τ on e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' First note that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) agrees set-wise with r−1 i Bn on e and at the endpoints of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The map ψ ◦ λ ◦ τ is length-multiplying (by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10(3) and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13(2)), log is length- multiplying on the circular segment ψ ◦ λ ◦ τ(e), and f is length-multiplying by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus the modulus of the derivative of f ◦log ◦ψ◦λ◦τ is constant on e, and so the derivatives of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) and r−1 i Bn have the same modulus at each point of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Conclusion (2) now follows from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Joining Different Types of Boundary Arcs: the Map En Recall that in Section 4 we defined the maps ηΨ i , ηi where 1 ≤ i ≤ ℓ, and in Section 5 we defined the map hn for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In this section we define a map En in D∗ which is roughly given by either z �→ ηΨ i ◦ hn(z) or z �→ ηi(z) ◦ hn(z), where i is allowed to depend on arg(z) and which of ηΨ i , ηi we post-compose hn with is also allowed to depend on arg(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, we will need a way to interpolate between the definitions of ηΨ i , ηi, for different i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The interpolation regions are defined in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 below, and the map En in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It will be useful to keep Figure 12 in mind for the remainder of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Mark one edge ei on Γi for each 1 ≤ i ≤ ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Label the ℓ components of ∂Ω′ n \\ ∪iei as (Gi)ℓ i=1, where ∂Di ⊂ Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let (1) J D i denote those edges in ∂Di, (2) J G i denote those edges in Gi \\ J D i , (3) J e denote the edges (ei)ℓ−1 i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In other words, J D i are the edges shared by ∂Ω′ n and ∂Di, J e consists of ℓ − 1 edges: one on each of the curves (Γi)ℓ i=1, and J G i are the remaining edges on Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus we have: ∂Ω′ n = J e ∪ � i � J D i ∪ J G i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For z ∈ D∗, we define: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) En(z) := � ηψ i ◦ hn(z) if z/|z| ∈ ψ ◦ λ ◦ τ(J D i ) ηi ◦ hn(z) if z/|z| ∈ ψ ◦ λ ◦ τ(J G i ) It remains to define En(z) for z ∈ D∗ satisfying z/|z| ∈ ψ ◦ λ ◦ τ(J e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We do so in the following Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The map En extends to a locally univalent K-quasiregular mapping En : D∗ → C satisfying En(z) = zm for |z| ≥ m√ 2, where m = m(n) is as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, K does not depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 22 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Illustrated is Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The curves Γ1, Γ2 are depicted as black dotted lines, except for the edges e1 ⊂ Γ1, e2 ⊂ Γ2 which are in thick black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Consider (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Note that if En is defined at z and |z| ≥ m√ 2, then En(z) = zm by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3(1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, setting En(z) := zm for |z| ≥ m√ 2 extends the definition of En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It remains to extend the definition of En to: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) {z : 1 ≤ |z| ≤ m√ 2 and z/|z| ∈ ψ ◦ λ ◦ τ(ei)}, for 1 ≤ i ≤ ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Each of the ℓ − 1 sets in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) consists of 2 quadrilaterals which we denote by Q± i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The curve Γi connects two distinct elements of (Di)ℓ−1 i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In order to avoid complicating notation significantly, we will assume without loss of generality that Γi connects Di to Di+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let γi ⊂ 2D be a smooth Jordan arc connecting ηΨ i (1) to ηΨ i+1(−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6), we can choose γi so that the union of the arcs (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) ηΨ i ([1, 2]), 2T ∩ H, ηΨ i+1([−2, −1]), γi forms a topological quadrilateral we denote by Q+ i (in particular none of the arcs in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) intersect except at common endpoints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Define a quasisymmetric homeomorphism g+ i : ∂Q+ i → ∂(A(1, 2) ∩ H) (see Figure 13) by g+ i (z) = z for z ∈ 2T g+ i (z) = (ηΨ i )−1(z) for z ∈ ηΨ i ([1, 2]) g+ i (z) = (ηΨ i+1)−1(z) for z ∈ ηΨ i+1([−2, −1]), and extending g+ i to a quasisymmetric homeomorphism of γi to T ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The mapping g+ i extends to a quasiconformal homeomorphism g+ i : Q+ i → A(1, 2) ∩ H (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='24 of D1 " G1A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 23 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Illustrated is the quadrilateral Q+ i and the map g+ i in the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' [BF14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We define En(z) := (g+ i )−1(zm) for z ∈ Q+ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A similar definition of g− i : Q− i → A(1, 2) ∩ −H is given (using the same curve γi) so that g+ i (z) = g− i (z) for z ∈ T ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We let En(z) := (g− i )−1(zm) for z ∈ Q− i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' To summarize, we have defined En in each of the three regions {z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(J D i )}, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) {z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(J G i )}, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) {z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(J e i )}, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) Indeed, the definition of En in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) was given already in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1), and in this proof we have defined En in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The definitions of En in each of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) agree along any common boundary, and thus by removability of analytic arcs for quasiregular mappings, it follows that En is quasiregular on D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, En has no branched points in D∗, and hence En is locally quasiconformal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The dilatation of the map En depends only on the dilatation of hn (which is independent of n by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3(3)) and the dilatations of the the finite collection of quasiconformal maps used in its definition: ηΨ i , ηi, g+ i , g− i , and hence we may take K independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Defining gn in Ω′ n First we recall our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We have fixed ε > 0, a compact set K, disjoint, analytic domains (Di)k i=1 so that K ⊂ U := ∪iDi, and f holomorphic in a neighborhood of U with ||f||U < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We defined curves {Γi}k−1 i=1 connecting the domains (Di)k i=1, and we denoted by Ω a component of the complement of U ∪ ∪k−1 i=1 Γi with τ : Ω → D∗ conformal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The domain Ω′ n is contained in Ω, and ψ ◦ λ ◦ τ maps Ω′ n onto D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In Section 6 we defined the map En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We define the mapping gn : Ω′ n → �C by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) gn := En ◦ ψ ◦ λ ◦ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Izl=2 Izl=2 Izl=1 Di D i+124 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK We will now record at which points the function gn|Ω′n is locally n : 1 for n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let g be a quasiregular function, defined in a neighborhood of a point z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We say that z is a branched point of g if for any sufficiently small neighborhood U of z, the map g|U is n : 1 onto its image for n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We say w ∈ C is a branched value of g if w = g(z) for a branched point z of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We denote the branched points of a quasiregular mapping g by BP(g), and the branched values by BV(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall that in Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 we fixed a point p ∈ Ω satisfying τ(p) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The mapping gn : Ω′ n → C of Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 is K-quasiregular and C- vertex supported for K, C independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, g−1 n (∞) = {p}, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) BP(gn) ⊂ � e∈∂Ωn {z : dist(z, e) < C · diam(e)}, and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) BV(gn) ⊂ k� i=1 Ψi(riT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since each of the mappings in the composition (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) are K-quasiregular and C-vertex supported for K, C independent of n, the same is true of gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The only points where the mapping gn is locally l : 1 for l > 1 are a subset of the vertices of the graph ∂Ω′ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13, the vertices of ∂Ω′ n all lie in � e∈∂Ωn {z : dist(z, e) < C · diam(e)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, any vertex of ∂Ω′ n is mapped to a point on one of the curves Ψi(riT) by gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Hence, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) follows since BV(gn) = gn(BP(gn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It remains to show: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) g−1 n (∞) = {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Indeed, note that En◦ψ◦λ fixes ∞ and has no finite poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The map τ : Ω → D∗ is conformal and hence only one point p is mapped to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The relation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) now follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ It will be useful to record the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then for all sufficiently large n, we have: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) gn(z) = τ(z)m for any z ∈ τ −1({z : |z| > r}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Consider the functional equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) defining gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The maps λ, ψ are vertex- supported, and moreover λ (respectively, ψ) is the identity outside of the support of λz, (respectively, ψz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7, we therefore have that ψ ◦ λ(z) = z if z ∈ τ −1({z : |z| > r}) and n is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The relation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) now follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3(1) since m → ∞ as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 25 Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' As in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13, we note that our Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 of gn is determined by a choice of the objects K, U, D, f, ε, Ω, p we fixed in Notations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' When we wish to emphasize this dependence, we will write gn(K, U, D, f, ε, Ω, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In particular, it will be useful in the next section to think of gn as a function taking as input any choice of K, U, D, f, ε, Ω, p satisfying the conditions in Notations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5, and outputting (via Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) a quasiregular function gn(K, U, D, f, ε, Ω, p) defined on Ω′ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Verifying gn is Quasiregular on �C In this section we combine our efforts in Sections 2-7 to define an approximant gn : �C → �C of a given f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The approximant gn will not be holomorphic as required in Theorems A and B, but we will solve this problem in the next section by applying the Measurable Riemann Mapping Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We fix the following for Sections 8-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Notation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Fix K, f, D, ε, P as in the statement of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Denote by U the neighborhood of K in which f is holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Define (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) P ′ := {p ∈ P : p is contained in a component V of �C \\ K such that V ̸⊆ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Compactness of K implies that U contains all but finitely many components of �C \\ K, and so the set P ′ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, P ′ does not depend on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By shrinking U if necessary, we may assume that: (1) U ∩ P ′ = ∅, (2) P ′ contains exactly one point in each component of �C \\ U, (3) f is holomorphic in a neighborhood of U ⊂ D, and (4) the components of U are a finite collection of analytic Jordan domains (Di)k i=1 so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) holds for each Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let K′ be a compact set such that K ⊂ int(K′) ⊂ K′ ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will assume for now that ||f||U < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We now define a quasiregular approximation gn of f by applying the Blaschke product construction of Section 2 in each Di, and by applying the folding construction of Sections 3-7 in each complementary component of ∪i(Di ∪ Γi): Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For every n, we define a quasiregular mapping gn as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recalling Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13, we first set (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) gn := gn(ε, K′ ∩ Di, Di, f|Di) in Di for 1 ≤ i ≤ k The equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) defines the curves (Γi)k i=1 by way of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3, and we enumerate the components of �C \\ � U ∪ k−1 � i=1 Γi � 26 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK by (Ω(i))ℓ i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recalling Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 and Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='14, we extend the definition of gn to the open set (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) Ω := �C \\ � ℓ� i=1 ∂Ω′ n(i) � by the formula (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) gn := gn(K′, U, D, f, ε, Ω(i), P ′ ∩ Ω(i)) in Ω′ n(i) for 1 ≤ i ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The quasiregular function gn is C-vertex supported and K-quasiregular for C, K independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For gn|Ω′n(i) this is exactly Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4, and so the conclusion follows since gn is holomorphic in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ The function gn is now defined on all of �C except for the edges of each ∂Ω′ n(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We show in Propositions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 below that gn in fact extends continuously across each edge of ∂Ω′ n(i), and deduce in Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 that gn extends quasiregularly across ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The K-quasiregular function gn : Ω → �C extends to a continuous function gn : Ω ∪ e → �C for any edge e ⊂ ∂Ω ∩ ∂U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let i be so that e ⊂ ∂Di and denote the unique element of (Ω(i))ℓ i=1 that contains e on its boundary by Ω(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall by Definitions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) gn|Di = Ψi ◦ Bn, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) gn|Ω′n(j) = En ◦ ψ ◦ λ ◦ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Assume that i ∈ I (the reasoning in the case i ̸∈ I will be the same), so that by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3(2) we have hn ◦ ψ ◦ λ ◦ τ = r−1 i Bn on e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) and the definition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) of En, it follows from (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) that gn|Ω′n(j)(z) = Ψi ◦ Bn(z) for z ∈ e, in other words gn|Ω′n(j) and gn|Di agree pointwise on e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The K-quasiregular function gn : Ω → �C extends to a continuous function gn : Ω ∪ e → �C for any edge e ⊂ ∂Ω ∩ (∪ℓ i=1Ω(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let j be so that e ⊂ ∂Ω′ n(j), and as in the proof of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4, recall that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) gn|Ω′n(j) = En ◦ ψ ◦ λ ◦ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let x ∈ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' There are two limits lim Ω′n(j)∋z→x ψ ◦ λ ◦ τ(z), A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 27 each lying on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Denote them by ζ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13(3), ζm + = ζm − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1), we conclude that there is a unique limit lim Ω′n(j)∋z→x En ◦ ψ ◦ λ ◦ τ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Hence, setting gn(x) := lim Ω′n(j)∋z→x En ◦ ψ ◦ λ ◦ τ(z) defines a continuous extension of gn across the edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The K-quasiregular function gn : Ω → �C extends to a K-quasiregular func- tion gn : �C → �C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The set �C \\ Ω = ∂Ω consists of a finite collection of analytic arcs: the edges of the graphs ∂Ω′ n(i) over 1 ≤ i ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, by removability of analytic arcs for quasiregular mappings, it suffices to show that gn : Ω → �C extends continuously across each such edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' There are two types of edges to check: those that lie on the boundary of a domain Di, and those that lie in the interior of a domain Ω(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We have already checked continuity across both types of edges in Propositions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5, and so the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof of the Main Theorems In Section 9 we prove Theorems A and B, modulo the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 which is left to Sections 10-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall that in Section 8 we fixed the objects K, f, D, ε, P as in Theorem B (see Notation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1), and we defined a quasiregular approximation gn to f in Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We also showed in Section 8 that gn in fact extends to a quasiregular function gn : �C → �C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Now we apply the MRMT below in Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 to obtain the rational maps rn : �C → �C which we will prove satisfy the conclusions of Theorems A and B for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The mapping gn induces a Beltrami coefficient µn := (gn)z/(gn)z, which, by way of the MRMT, defines a quasiconformal mapping φn : �C → �C such that rn := gn ◦ φ−1 n is holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We normalize φn so that φn(∞) = ∞ and φn(z) = z + O(1/|z|) as z → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We now begin deducing that for large n, the maps rn satisfy the various conclusions in Theorems A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The function rn of Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 is rational, and r−1 n (∞) = φn(P ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In particular, if K is full and P = {∞}, then rn is a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The function rn is holomorphic on �C and takes values in �C: the only such functions are rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Note that g−1 n (∞)∩U = ∅ since gn is bounded on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4 and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4), we have that g−1 n (∞) = P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since rn := gn ◦ φ−1 n , we conclude that r−1 n (∞) = φn(P ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 28 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK The last statement of the proposition follows since we normalized φn(∞) = ∞, and the only rational functions with a unique pole at ∞ are polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For all R < ∞, the mapping φn satisfies: (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) ||φn(z) − z||R·D n→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since gn is C-vertex supported by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3, we conclude from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7 that (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) Area(supp(µn)) n→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The relation (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) now follows from (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) since ||µn||L∞ ≤ K for all n by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For all sufficiently large n, the mapping rn satisfies CP(rn) ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7, we have max � diam(e) : e is an edge of ℓ� i=1 ∂Ω(i) � n→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, since D is a domain containing ∪ℓ i=1∂Ω(i), we have by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) that BP(gn) \\ U ⊂ D for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since U ⊂ D we conclude that BP(gn) ⊂ D for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The result now follows from Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 since φ(BP(gn)) = CP(rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For all sufficiently large n, we have ||rn − f||K < 3ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' First we note that since f is uniformly continuous on K′, there exists δ > 0 so that if z, w ∈ K′ and |z − w| < δ, then |f(z) − f(w)| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3, we can conclude that (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) ||φn(z) − z||K < min(δ, dist(K, ∂K′)) for all sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let z ∈ K, w := φ−1 n (z) and suppose j is such that z ∈ Dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then |rn(z) − gn(z)| = |gn(w) − gn(z)| ≤ |gn(w) − f(w)| + |f(w) − f(z)| + |f(z) − gn(z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) Let C := sup z∈D 1≤i≤k |Ψ′ i(z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then ||gn − f||K′ ≤ C · ||Ψ−1 j gn − Ψ−1 j f||K′, and we deduce by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='15) that: ||gn − f||K′ ≤ C · ||Bn − Ψ−1 j f||K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 29 Applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10 (see also Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9), we conclude that (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) ||gn − f||K′ n→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Next, we deduce from (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) that |rn(z) − gn(z)| < 2ε for sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It follows that for sufficiently large n: ||rn − f||K ≤ ||rn − gn||K + ||gn − f||K < 3ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For all sufficiently large n we have CV(f|K) ⊂ CV(rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let z ∈ CP(f|K), and let i be so that z ∈ Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then Ψ−1 i ◦f(z) ∈ CV(Bn) by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus f(z) ∈ CV(Ψi ◦ Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, by the Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12 of gn, we have for large n that f(z) ∈ BV(gn) = CV(rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For all sufficiently large n, we have CV(rn) ⊂ fill{z : d(z, f(K)) < ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since CV(rn) = BV(gn), it suffices to show that for every z ∈ BP(gn) and sufficiently large n, we have (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) gn(z) ∈ fill{z : d(z, f(K)) < ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For z ∈ BP(gn) \\ U, (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For z ∈ BP(gn) ∩ Di, (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) follows from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12 of gn|Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Proof of Theorem B: In the special case that ||f||K < 1, we have already proven that the mappings rn satisfy the conclusions of Theorem B for all sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Indeed, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 says that ||rn − f||K < ε, conclusion (2) in Theorem B is Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4, and conclusion (3) is Theorems 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Conclusion (1) follows from Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The general case follows by applying the above special case to an appropriately rescaled f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Proof of Theorem A: When K is full, we may take P = {∞} and apply Theorem B, in which case Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 guarantees that the maps rn are polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (Mergelyan+) Let K ⊂ C be full, suppose f ∈ C(K) is holomorphic in int(K), and let D be a domain containing K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For every ε > 0, there exists a polynomial p so that ||p − f||K < ε and: (1) CP(p) ⊂ D, (2) CV(p) ⊂ fill{z : d(z, f(K)) ≤ ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 30 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Proof : By the usual version of Mergelyan’s Theorem, there exists a polynomial q so that ||q − f||K < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Apply Theorem A to K, D, q, ε/2 to obtain an approximant of q which we denote by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The polynomial p satisfies the conclusions of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (Weierstrass+) Suppose that I ⊂ R is a closed interval, f : I → R is continuous, and U, V ⊂ C are planar domains containing I, f(I), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then, for every ε > 0, there exists a polynomial p with real coefficients so that ∥f − p∥I ≤ ε, and (1) CP(p) ⊂ U, (2) CV(p) ⊂ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let I = [a, b], and f, U, V as in the statement of the corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8, there exists a complex polynomial q so that ||q − f||[a,b] < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The real polynomial Q(z) := q(z) + q(z) 2 satisfies Q(x) = Re(q(x)) for x ∈ R and hence ||Q − f||[a,b] < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will use the symbol ⋐ to mean compactly contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let V1 ⋐ V be a sufficiently small, R-symmetric domain containing f(I) so that there is a component of Q−1(V1) (which we denote by U1) satisfying U1 ⋐ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let U2 be a R-symmetric, analytic domain satisfying I ⋐ U1 ⋐ U2 ⋐ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall Notation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 and consider: (1) the compact set U1, (2) the analytic function Q, (3) the analytic domain U2 containing U1, (4) min{ε/2, dist(∂V1, ∂V )}, (5) P = {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Applying Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 to (1)-(5) yields quasiregular mappings gn with R-symmetric Bel- trami coefficient, so that (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) pn := gn ◦ φ−1 n is a real polynomial approximant of Q for large n satisfying: (1) ||pn − f||[a,b] < ε, (2) CP(pn) ⊂ U2, (3) CV(pn) ⊂ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus pn satisfies the conclusion of Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9 for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If we make further assumptions on the compact set K, the conclusion CV(r) ⊂ fill{z : d(z, f(K)) < ε} of Theorems A and B can be improved to (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8) CV(r) ⊂ fill(f(K)), A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 31 which is equivalent to CV(r) ⊂ f(K) if f(K) is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Indeed, if for instance the interiors of K, f(K) are analytic domains and f : int(K) → int(f(K)) is proper, then a similar strategy as in the proofs of Theorems A and B but replacing Ψ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='15) with a conformal map D �→ int(K) can be used to prove (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall the notation Ω(i) from Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2, and let τi : Ω(i) → D∗ be the conformal mapping satisfying τ −1 i (∞) = P ′ ∩ Ω(i) as in Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The following fact justifies part of our description in the introduction of the behavior of the rational approximants off K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let 1 < r < R < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then, for all sufficiently large n, we have (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9) rn ◦ φn(z) = τi(z)m and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10) |rn(z)| > R for all z ∈ τ −1 i ({z : |z| > r}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Fix R < ∞ and r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' From (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) and the functional equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) defining gn in Ω(i), it follows that: (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11) gn(z) = τi(z)m for all z ∈ τ −1 i ({z : |z| > (r + 1)/2}) for all large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12) rn ◦ φn = gn, The relation (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, we have by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 that: (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13) φn ◦ τ −1 i ({z : |z| > r}) ⊂ τ −1 i ({z : |z| > (r + 1)/2}) for all sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since ((r + 1)/2)m > R for large n, the relation (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10) also follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Some Estimates Involving Harmonic Measure and Green’s Functions In Sections 10-12 we turn to the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In fact, we will prove a slightly stronger result (Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 in Section 12) from which Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We begin by recalling a few standard facts, and sketch the proofs for the convenience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let D(z, r) denote the disk of radius r centered at z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 is illustrated in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If K ⊂ D is continuum connecting 0 to T, then ω(z, K, D \\ K) ≥ c > 0 for all |z| < 1/4 and some c > 0 independent of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 32 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Illustrated is the compact set K of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Exercise III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10 of [GM08] says that if E is a continuum connecting {|z| = 1 2} to T in D, then ω(0, E, D \\ E) ≥ c = 2 π tan−1(1/ √ 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Apply the exercise and the maximum principle to the disk D = D(z, 1 2) to deduce the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' [GM08] gives a simple direct proof of the exercise, but it also follows from Beurling’s projection theorem, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 of [GM08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose u, v are harmonic functions on D such that supD |u|, supD |v| ≤ M and that |u − v| < ε on some continuum K connecting 0 to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then |u − v| ≤ εcM 1−c on D(0, 1/4), where c > 0 is the constant from Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Consider the subharmonic function log |u−v|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It is less than log ε on K and less than log M on ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus for |z| ≤ 1/4, Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 implies log |u(z) − v(z)| ≤ ω(z, K, D \\ K) log ε + ω(z, T, D \\ K) log M ≤ c log ε + (1 − c) log M, so |u(z) − v(z)| ≤ εcM 1−c, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose Ω is a planar domain, K ⊂ Ω is compact and connected, and 0 < ε, M < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then there is a δ > 0 so that if h = u + i�u is holomorphic on Ω, �u vanishes at some point of K, supΩ |h| ≤ M and supK |u| < δ, then supK |�u| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If K is single point, this is trivial since �u = 0 there by assumption, so assume K is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Choose η > 0 so that η < dist(K, ∂Ω) and η < diameter(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then for any radius η disk D centered at a point of K, u is less than δ on a continuum connecting the center of D to its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This implies |u| ≤ δcM 1−c (for c as in Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) on a η/4-neighborhood of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus |∇u| = O(δcM 1−c/η) on the (η/8)-neighborhood U of K (this uses the Cauchy estimate for |∇u|, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4 of [ABR01]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since |∇�u| = |∇u| and Iwl=1 K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='0 Z wl=1/4A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 33 Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Illustrated in gray is the set Uδ of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' U is path connected, this implies �u is within ε of zero on K if δ is small enough (depending on ε, η, M and the diameter of U in the path metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ We will use this later in the situation that if u and v are harmonic functions on Ω that are close enough on K ⊂ Ω, then f = exp(u + i�u) and g = exp(v + i�v) are holomorphic functions on Ω that are close on K (if �v is chosen to agree with �u at some point of K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Next, we recall the well known boundary Harnack inequality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', see Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='18 of [Mar19] or Exercise I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 of [GM08]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose u and v are positive harmonic functions on D which extend contin- uously to the boundary T, and suppose furthermore that u and v are both equal to zero on an arc I ⊂ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For δ > 0 let Uδ = {z ∈ D : dist(z, T \\ I) > δ} (see Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then for z ∈ Uδ, δ2 4 · u(0) v(0) ≤ u(z) v(z) ≤ 4 δ2 · u(0) v(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Note that, under these conditions, u and v have well defined inward normal derivatives on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By letting z → T radially, the inequalities above imply that ∂u ∂n and ∂v ∂n are comparable (with the same constants as above) at any point of Uδ ∩ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose u is a positive harmonic function on D which extends continuously to the boundary T and that equals zero on an arc I ⊂ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose a, b ∈ I both have distance > δ from T \\ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then δ2 4 · ∂u ∂n(a) ≤ ∂u ∂n(b) ≤ 4 δ2 · ∂u ∂n(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We simply compare u to a rotation of itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let v(z) = u( b az).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then v and u both vanish on J = I ∩ a b · I, and a ∈ J is distance > δ from either endpoint of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Hence the normal derivatives of u and v at a are comparable by the boundary Harnack principle, and hence so are the normal derivatives of u at a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Us 134 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Suppose Ω is an analytic domain (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For w ∈ Ω, let G(z, w) be the Green’s function on Ω with pole at w, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', G is harmonic on Ω \\ {w}, vanishes identically on ∂Ω and G(z, w) + log |z − w| is bounded in a neighborhood of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Our assumptions on Ω imply that Ω is regular for the Dirichlet problem, and hence that the Green’s function exists and is unique for any w ∈ Ω (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', see Sections II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 of [GM08]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose Ω is an analytic domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If r > 0 is small enough (depending only on Ω), x ∈ ∂Ω, and y ∈ Ω \\ D(x, r), then the normal derivative of the Green’s function with pole at y has comparable size at all points of ∂Ω ∩ D(x, r/2) with a constant independent of y and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Choose r small enough that W = D(x, r) ∩ Ω is a Jordan domain whose boundary consists of a sub-arc γ of ∂Ω and an arc of the circle ∂D(x, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let �γ = ∂Ω ∩ D(x, r/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Choose a point z ∈ W that is about distance r from ∂W and choose a conformal map ϕ : W → D taking z to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If r is small enough, ϕ extends analytically across γ to all of D(x, r) and by the Koebe distortion theorem it has comparable derivative at all points of �γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Also, since γ and each component of γ \\ �γ has harmonic measure with respect to z that is bounded away from zero, the image arcs on T all have lengths bounded uniformly from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus u(z) = G(ϕ−1(z), y) is a positive harmonic function on D vanishing on ϕ(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 the normal derivatives of u are comparable at all points of ϕ(�γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since the values of |ϕ′| are comparable at all points of �γ, we can deduce the lemma from the chain rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose Ω is an analytic domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If r > 0 is small enough (depending only on Ω), x ∈ ∂Ω, and y ∈ Ω, then ∂G(x, y)/∂n is comparable at all points of γ = ∂Ω ∩ D(x, r) with a constant depending only on an upper bound for M = max � 1, ℓ(γ) dist(y, ∂Ω) � , where ℓ(γ) denotes the length of γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, ∂G(x, y)/∂n = O(1/r) on γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We can cover γ by at most O(M) disks Dj = D(xj, rj) whose doubles are all disjoint from y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 implies the normal derivatives are comparable with some uniform constant C on each corresponding arc, so they are comparable with constant CO(M) on γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (In fact, if r is so small that ∂Ω looks “straight” on scale r, then only O(log M) disks are needed, since they become geometrically larger as we move away from y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=') The final claim follows because the integral of the normal derivative over the whole bound- ary is 2π, and hence the integral over γ is ≤ 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since the size of the normal derivative is comparable at all points of γ, this implies it is bounded above by O(1/ℓ(γ)) = O(1/r) (recall ℓ denotes length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ The following two lemmas relate harmonic measure to Green’s function: we refer to Figure 16 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 35 (a) (b) Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In (A) the setup for Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8 is shown, and in (B) the setup for Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' There is a constant C1 < ∞ so the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose Ω is an analytic domain, that Γ is a connected component of ∂Ω and that w ∈ Ω satisfies dist(w, Γ) = dist(w, ∂Ω) ≤ diameter(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then G(z, w) ≤ C1 on σ = {z : |z − w| = 1 2 dist(w, Γ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let Ω′ be the component of �C \\ Γ containing Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then Ω ⊂ Ω′, so by the maximum principle the Green’s function for Ω is less than the Green’s function for Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus it suffices to prove the lemma for the simply connected domain Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' But by Koebe’s distortion theorem, σ contains a ball of fixed hyperbolic radius around w and hence its image contains a ball of fixed radius if we conformally map Ω′ to the disk with w going to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' On the disk, the Green’s function is log 1 |z| which is clearly bounded by some C outside a fixed ball around the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' There is a constant C2 < ∞ so that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose that Ω is an analytic domain, z ∈ Ω, that γ ⊂ ∂Ω is a subarc, and that w ∈ Ω satisfies (1) dist(w, Γ) = dist(w, ∂Ω) ≤ diameter(Γ), where Γ is the component of ∂Ω containing γ, (2) |z − w| ≥ dist(w, ∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then G(z, w) ≤ C2ω(z, γ, Ω)/ω(w, γ, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let C1 and σ = {z : |z − w| = 1 2 dist(w, Γ)} be as in Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By assumption z is in the component Ω′ of Ω \\ σ not containing w and by the maximum principle applied to Ω′, ω(z, σ, Ω′) ≥ G(z, w)/C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By the maximum principle (again applied to Ω′), ω(z, γ, Ω) ≥ ω(z, σ, Ω′) · min x∈σ ω(x, γ, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' r 5 Wr Z W 5 Y36 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK By Harnack’s inequality (see Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='17 of [Mar19]), all the values of ω(x, γ, Ω) are comparable on σ and hence there is an ε > 0 so that ω(z, γ, Ω) ≥ ω(z, σ, Ω′) · ε · ω(w, γ, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Finally, by Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8 we have ω(z, γ, Ω) ≥ (ε/C1) · G(z, w) · ω(w, γ, Ω), which is the desired inequality with C2 = C1/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose Ω is an analytic domain, z ∈ Ω and ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose also that {γk}n 1 ⊂ ∂Ω is a collection of disjoint arcs and {wk}n 1 ⊂ Ω is a collection of points, so that for all k we have: (1) dist(wk, Γk) = dist(wk, ∂Ω) ≤ diameter(Γk), where Γk is the component of ∂Ω con- taining γk, (2) |z − wk| ≥ dist(wk, ∂Ω), (3) ω(wk, γk, Ω) ≥ ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then �n k=1 G(z, wk) ≤ C2/ε where C2 is the constant from Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9, � k G(z, wk) ≤ (C2/ε) � k ω(z, γk, Ω) and since the arcs {γk} are disjoint, we have � k ω(z, γk, Ω) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Periods of Harmonic Functions Suppose Ω is an analytic domain with N +1 boundary components Γ0, Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' , ΓN, so that Ω is regular for the Dirichlet problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose h is harmonic in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In any sub-disk D ⊂ Ω, h has a harmonic conjugate �h that is well defined up to an additive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If γ is a closed curve in Ω, then we can analytically continue �h along γ until we return to the starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The period of h along γ is the difference between the starting and ending values of �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If γ is homologous to a point, the period is zero, but if γ is homologous to a boundary component of Ω, then the period may be non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The sum of the periods corresponding to all N + 1 boundary components is always zero (the union of all boundary curves is homologous to zero in Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Next we consider the periods of certain special functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' , N let ωj be the harmonic function on Ω that has boundary value 1 on Γj and is 0 on the other boundary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since the boundary components are analytic, each ωj extends to be analytic across ∂Ω, so the normal and tangential derivatives are well defined, and themselves analytic, at every boundary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The period of ωj along Γk is the integral of the tangential derivative of �ωj around Γk, and this equals the integral of the normal derivative of ωj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', the period is λjk = � Γk ∂ωj ∂n ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 37 Note that since 0 < ωj < 1 in Ω and ωj = 1 on Γj, the inward normal derivative of ωj on Γj is non-positive (and strictly negative by analyticity: in that case, G can’t have critical point on the boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Similarly, the inward normal derivative of ωj is strictly positive on Γk for k ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus λjj < 0 and λjk > 0 for k ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since the periods of ωj sum to zero we have N � k=0 λjk = 0 for every j, and since � j ωj is the constant function 1 on Ω, we have N � j=0 λjk = 0 for every k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus for each j we have: λjj = − � k≥0:k̸=j λjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In other words, the row and column sums are all zero for the (N +1)×(N +1) matrix (λjk), 0 ≤ j, k ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If we drop the first row and column of this matrix, we are removing all positive terms (except for λ00), so we have j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' , N, λjj < − � k≥1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='k̸=j λjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus the N × N matrix Λ = (λjk), 1 ≤ j, k ≤ N is diagonally dominant, and this implies it is invertible by the Levy-Desplanques theorem: if the kernel of Λ contains a non-zero vector v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' , vN), then � k≥1 vkλjk = 0, and if |vk| is the largest component of v, then |vkλkk| = | � j≥1:j̸=k λjkvj| ≤ |vk| · � j≥1:j̸=k |λjk| < |vk| · |λkk|, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' See Olga Taussky-Todd’s paper [Tau49] for some history of this oft-rediscovered fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Also note that ∥Λ∥ and ∥Λ−1∥ only depend on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We have now proven the following result (due in a slightly different form to Heins [Hei50] and in greater generality to Khavinson [Kha84]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose Ω, Λ and {ωj}N 1 are as above and suppose we assign real numbers v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' , vN) to the N boundary components Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' , ΓN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then there is a linear combination h = �n j=1 ajωj so that the period of h around Γj is exactly vj for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The coefficients a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' , aN) are solutions of the linear equation Λa = v and hence ∥a∥ ≤ ∥v∥ · ∥Λ−1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus if ∥v∥ ≤ ε then ∥a∥ = O(ε) with a constant that depends only on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We say that the periods of a harmonic function h on Ω are well defined modulo 2π if every period is some integer multiple of 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In this case, f = exp(h + i�h) is a well defined 38 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK analytic function on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For example, if Ω is the complement of a finite set of points {zk}n 1, then �n k=1 log |z − zn| has periods that are well defined modulo 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The corresponding holomorphic function is a polynomial with zeros at {zk} (and hence extends holomorphically from Ω to the whole plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Note that ∇�u is always well defined even if �u is not, since any two different branches of �u differ by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose Ω is an analytic domain and K ⊂ Ω is a compact set that contains curves {σj}N 0 homologous to each of the boundary components {Γj}N 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose K ⊂ W ⊂ Ω is open, let η = dist(K, ∂W) and set U = {z ∈ W : dist(z, ∂W) > η/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose P ⊂ Ω is a finite set and suppose u and H are harmonic functions on Ω \\ P, and each is either bounded or has a logarithmic pole at each point of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose |u − H| is bounded by M on U, and that |u − H| < ε on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If u has a well defined harmonic conjugate modulo 2π on Ω, then there is an harmonic h on Ω so that (1) h + H also has a well defined harmonic conjugate modulo 2π on Ω \\ Z, (2) |h| ≤ Cεc on all of Ω (not just K), (3) h is constant on each component of ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The constant c is the same as in Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 and C depends only on Ω and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since v = u − H is bounded on U, it extends to be harmonic at each point of P ∩ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since |v| < ε on K, it is bounded by O(εcM 1−c) on U and hence the gradient of v is bounded by O(εcM 1−c/η) on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus the gradient of the (possibly multi-valued) harmonic conjugate of v is bounded by the same quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We deduce that the harmonic conjugates of H and u differ by δ = O(LεcM 1−c/η) on U, where L is the diameter of U in the path metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus the periods of H on the {σj} differ from multiples of 2π by at most δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Now apply Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 to define a harmonic function h on Ω that is bounded by O(δ) and has exactly the periods of −v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The Generalized Caratheodory Theorem on Blaschke Approximation If B is a finite Blaschke product (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) on an analytic domain Ω, then it has non-zero, continuous boundary values, and hence can have only finitely many zeros inside Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, the Schwarz reflection principle implies B extends holomorphically across ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The following result extends Carath´eodory’s Theorem (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1), and its statement uses the notation IB (see Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4), and the following notation: Notation 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If Ω ⊂ C is an analytic domain and Γ := ∂Ω, we denote Γδ := {z ∈ Ω : dist(z, Γ) = δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose that Ω ⊂ C is an analytic domain, K ⊂ Ω is compact, and f is holomorphic on a neighborhood of Ω with supΩ |f| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let Γ := ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then for any ε > 0 and sufficiently small δ > 0, there is a finite Blaschke product B on Ω so that (1) supK |f − B| ≤ ε, A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 39 (2) 1 − ε < |B| ≤ 1 on ∂Ω, (3) Every component of Γδ \\ {B = 0} has length comparable to δ and adjacent connected components have length comparable to within a factor of 1 + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (4) All components of IB have length which is comparable to δ with constants that depend only on ε and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (5) on each component γ of IB, the ratio maxγ |B′|/ minγ |B′| is bounded depending only on ε and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Without loss of generality, we may assume K is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Otherwise, replace K by a compact, connected superset, for instance, its closed convex hull in the hyperbolic metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' As before, let G(z, w) denote the Green’s function on Ω with pole at w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since f is holomorphic on a neighborhood of Ω, it only has finitely many zeros in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We consider g := (1 − a) · f + b for some constants 0 < a < ε and |b| < ε such that |g| < 1 − ε on Ω and g has no zeros on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If we construct a finite Blaschke product B approximating g to within ε on K, then it approximates f to within 3ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Fix a finite number of smooth curves {σk} that are homologous to each boundary curve of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By enlarging K, if necessary, we may assume it contains all these curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let {zk}N 1 be the zeros of g, counted with multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By enlarging K again, if necessary, we may assume all the zeros of g are in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let W be an open domain with K ⊂ W ⊂ W ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By compactness we have min∂W |g| ≥ η for some η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In what follows, δ > 0 will always be chosen so small that W is disjoint from {z ∈ Ω : dist(z, ∂Ω) < 2δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In particular, g has no zeros in this neighborhood of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let u(z) = − log |g(z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then u ≥ − log(1 − ε) ≥ ε is positive and harmonic on Ω except for finitely many logarithmic poles at the {zk}N 1 , the zeros of g listed with multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let p(z) = �N k=1 G(z, zk) be the sum of the Green’s functions with these poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then v(z) = u(z)−p(z) is harmonic on Ω, and equals u on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus v is continuous and non-zero on ∂Ω, and hence it is bounded and bounded away from zero there, say m ≤ v ≤ M on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Hence m ≤ v ≤ M on all of Ω by the maximum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 of [GM08] v is the Poisson integral of its boundary values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', v(z) = 1 2π � ∂Ω v(w)∂G(w, z) ∂n ds(w) where ∂ ∂n is the inward normal and ds denotes length measure on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For w ∈ Γδ, denote by w∗ ∈ ∂Ω the closest point to w on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For z ∈ K and w ∈ Γδ we have G(w, z) = δ · ∂G(w∗, z) ∂n + O(δ2), where the constant depends on z, but is uniformly bounded as long as z is in the compact set K (the constant in the “big-Oh” depends on a bound for |∇2G| between Γδ and ∂Ω and since G is harmonic and extends analytically across ∂Ω, this is bounded as long as the pole 40 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK of Green’s function is not too close to ∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It follows that v(z) = 1 2πδ � Γδ v(w)G(w, z)ds(w) + O(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) Next use the identity G(z, w) = G(w, z) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8 of [GM08]), to deduce v(z) = 1 2πδ � Γδ v(w)G(z, w)ds(w) + O(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We now discretize the integral by cutting Γδ into disjoint subarcs {γk} chosen so that 1 ≤ � γk v(w) 2πδ ds(w) ≤ 1 + O(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) This is possible since the integral over a component Γk δ of Γδ is at least A = m · ℓ(Γk δ)/2πδ and this tends to infinity as δ ↘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We can therefore cut each boundary curve into sub-arcs where the integral is between 1 and 1+O(1/A) = 1+O(δ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', we can make the sub-integrals all as close to 1 as we wish, by taking δ small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The left side of Equation (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) implies that for each k, we have ℓ(γk)M/2πδ ≥ � γk v/2π ≥ 1 and hence ℓ(γk) ≥ 2πδ/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Similarly, the other side implies ℓ(γk) ≤ (1 + O(δ))2πδ/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus every such arc has length comparable to δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If δ is small enough, then the continuity implies that on the union of two adjacent arcs with common endpoint x, v is close to v(x), and hence by (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2), the lengths of these intervals are both close to 2πδ/v(x), and hence are close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This fact, together with each γk have length comparable to δ, will imply part (3) of the Theorem once we have defined the generalized Blaschke product B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Adding and subtracting a term from (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1), have v(z) = 1 2πδ � j G(z, wj) � γj v(w)ds(w) (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) + 1 2πδ � j � γj v(w)[G(z, w) − G(z, wj)]ds(w) + O(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The curve Γδ is parallel to the boundary, which is the level line G(z, w) = 0 of the Green’s function with pole at w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, the gradient of G along Γδ is nearly perpendicular to Γδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Denoting by wj the center of γj, we conclude: |G(z, w) − G(z, wj)| = O(δ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) Using (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) in the last term of Equation 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 gives v(z) = 1 2πδ � j G(z, wj) � γj v(w)ds(w) + � j 1 2πδ � γj v(w)O(δ2)ds(w) + O(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 41 Simplifying, we get v(z) = 1 2πδ � j G(z, wj) � γj v(w)ds(w) + O(δ) � j � γj v(w)ds(w) + O(δ) = � j G(z, wj) � γj v(w) 2πδ ds(w) + O(δ) = � j G(z, wj)(1 + O(δ)) + O(δ), = � j G(z, wj) + O(δ), where in the last line we have used Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10 to bound the sum of Green’s functions by O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Therefore, v is approximated on K by a finite sum of Green’s functions on Ω (indeed, we can even take the approximation to hold on the larger compact set W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If Ω is simply connected, then we are essentially done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In this case, H(z) = � k G(z, zk) + � j G(z, wj) = p(z) + � j G(z, wj) is harmonic except for a finite number of logarithmic poles at P = {∪kzk}∪{∪jwj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Therefore H has a harmonic conjugate �H that is well defined modulo 2π on Ω \\ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then B(z) = exp(−H − i �H) is holomorphic on Ω \\ P, but tends to zero at each point of P, so B is holomorphic on all of Ω with zeros exactly at the points of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, on ∂Ω we have |B| = exp(0) = 1, so B is a finite Blaschke product on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, H(z) = p(z) + � j G(z, wj) = u(z) − v(z) + � j G(z, wj) ≈ u(z) = − log |g(z)|, so log |B| = −H approximates log |g| on W as closely as wish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In particular, we may assume |B| ≥ η/2 on ∂W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In this case, g/B is holomorphic on W (since every zero of B inside W is also a zero of g of the same multiplicity), and so by the maximum principle |g/B| is bounded on K by max∂K |g/B| ≤ maxK 2|g|/η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Now apply Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 to h = log g/B = u+ �u on W to deduce that arg(B) approximates arg(g) on K, at least if we add an appropriate constant to arg(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Therefore B (times an appropriate unit scalar) uniformly approximates g on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This extends Carath´eodory’s theorem to simply connected domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' However, if Ω is multiply connected, then H need not have a well defined harmonic con- jugate modulo 2π on Ω \\ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Therefore B = exp(−H − i �H) need not be well defined: if we analytically continue B along one of the closed loops σk, we return to the same absolute value, but possibly a different value of the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The change in the argument is as small as we wish, tending to zero as the difference between log |B| = H and log |g| tends to zero 42 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This is because g has a well defined harmonic conjugate modulo 2π on Ω \\ P, and so the discussion following Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In order to get a well defined (generalized) finite Blaschke product B on Ω, we apply Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 to u and H to construct h so that (1) h is harmonic on all of Ω, and (2) h + H has a well defined harmonic conjugate modulo 2π on Ω \\ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (Note the H was constructed exactly so that the corollary can be applied: u−H is harmonic except for poles on Γδ which are outside W, and we may make u − H is as close to zero on K as we wish, while keeping it uniformly bounded on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=') Now set F = −H − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This function has all periods equal to zero modulo 2π, so B = exp(F + i �F) is a well defined holomorphic function on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since |h| = O(δ), we can deduce that B still approximates g on the compact set K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, since H = 0 on ∂Ω and h = aj on Γj, we see that |B| = exp(aj) = 1 + O(δ) on Γj, so |B| is constant on each boundary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Dividing by the largest such value, we get another finite Blaschke product that satisfies (2) and still approximates g to within O(δ) on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' To prove (4), it suffices to show that the modulus of the tangential derivative of B along ∂Ω (which is equal to |B′| since B is holomorphic) is comparable to 1/δ everywhere: then the preimage of a half-circle will have length comparable to δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Note that on ∂Ω, |B′| is also equal to the normal derivative of h + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The function h is a linear combination of fixed functions ωj that depend only on Ω, and although the coefficients of the combination may depend on δ, they remain small as δ ↘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus the normal derivative of h remains bounded on ∂Ω as δ ↘ 0, and this is negligible compared to 1/δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The function H is a sum of two sets of Green’s functions, one with poles {zj} corresponding to the zeros of g and the other with poles {wk} along the curve Γδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The first set of poles is fixed independent of δ and their contribution to the normal derivative of H is also bounded independent of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Again, these terms are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The main contribution to the normal derivative of H comes from the poles {wk} lying on Γδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For each such point wk consider the arc σk = D(wk, 2δ) ∩ ∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' the harmonic measure of this arc with respect to wk is bounded uniformly away from zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', ω(wk, σk, Ω) > c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This harmonic measure is the integral over σk of the normal derivative of the Green’s function with pole at wk, and thus it is less than the integral of the normal derivative of H, since this Green’s function is one term of the sum defining H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus the integral of |B′| over σk is > c and so any single arc lj in IB can contain at most a bounded number of arcs of the form σk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Part (3) of the theorem, adjacent points wk are at most distance O(δ) apart and hence lj can have length at most O(δ) (otherwise it would cover too many of the arcs σk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Next, we need to prove ℓ(lj) is bounded below by a multiple of δ, and this is equivalent to proving an upper bound |B′(x)| = O(1/δ) for any x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' As noted above, the contributions to |B′| from h and from the poles of H coming from the zeros of g are both bounded independent of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' To deal with the poles {wk} of H on Γk, we choose a point z ∈ Γδ that is A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 43 close to x ∈ ∂Ω, say |x − z| ≤ 2δ and note that ∂ ∂nG(x, wk) ≃ 1 δG(z, wk) Thus by Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10, the total contribution of all the poles {wk} to |B′| is at most � k 1 δG(wk, z) = O(1 δ), and hence (4) is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let γ denote a component of IB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7 and conclusion (4) just proven, the normal derivative of Green’s function with a pole at least distance δ from ∂Ω has comparable values at all points of γ, and so the same holds for any finite sum of such functions (with the same constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since on ∂Ω we have |B′| = | � ∂G ∂n |, we deduce (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If Ω = D, then it suffices to assume f is holomorphic on Ω instead of a neighborhood of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In that case if r < 1 then g(z) = f(rz) is holomorphic on a neighborhood of D and approximates f on a compact set K ⊂ D if r is close enough to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' So if B is a finite Blaschke product on Ω approximating g to within ε/2, and r is chosen so that g approximates f to within ε/2, then B approximates f to within ε, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The analyticity of f on a neighborhood of Ω is only used to deduce that f has a finite number of zeros inside Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The same proof would work if we assumed that f is holomorphic on Ω, extends continuously to ∂Ω, and is non-zero on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If we make the previous assumption on f, then it suffices to assume Ω is bounded and finitely connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If so, and any component of ∂Ω is a single point p, then by the Riemann removable singularity theorem, f extends to be holomorphic at p, and it suffices the prove the theorem for the extended function on the domain Ω′ = Ω ∪ {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By removing all the point components of ∂Ω, we may assume every component of Ω is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By repeated applications of the Riemann mapping theorem to the complement of each complementary component of Ω, the domain Ω can be mapped to a domain Ω′ bounded by a finite number of analytic curves indeed, with more work, the Koebe circle domain theorem says it can be mapped to a domain bounded by circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Transferring f and K to Ω we can use Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 to construct an approximating finite Blaschke product on Ω′, and then transfer this to a finite Blaschke on Ω that approximates f on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We placed the poles of our Green’s function all at the same distance δ from ∂Ω, but this was not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If we fix z0 ∈ Ω and let w ∈ Ω approach x ∈ ∂Ω, then G(z, w)/G(z0) approaches a positive harmonic function on Ω with zero boundary values on ∂Ω, except at x, where it blows up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus the limiting function must be a multiple of the Poisson kernel on Ω with respect to x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus it is easy to find finite weighted sums of Green’s functions (with positive real weights) that approximate the Poisson integral of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 44 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Then one must cluster the poles to approximate this sum by a sum of Green’s functions with integral weights;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' exponentiating such a sum gives a finite Blaschke product on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Possibly this extra flexibility would be useful in other problems, such trying to minimize the number of poles needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6: The hypotheses of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 are stronger than those of Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2: namely we assume in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 that ||f||Ω < 1 (rather than ≤ 1) and that the zeros of f are disjoint from ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Under these additional assumptions, there is no need in the second paragraph of the proof of Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 to replace f by g := (1 − a) · f + b since we already have |f| < 1 − ε and f has no zeros on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since the B produced in Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 approximates g to within O(δ) and we may take δ → 0, the conclusions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 follow from the conclusions of Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ References [ABR01] Sheldon Axler, Paul Bourdon, and Wade Ramey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Harmonic function theory, volume 137 of Grad- uate Texts in Mathematics.' metadata={'source': 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Bishop, Mathematics Department, Stony Brook University, Stony Brook, NY 11794- 3651 Email address: bishop@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='edu Kirill Lazebnik, Mathematics Department, University of North Texas, Denton, TX, 76205 Email address: Kirill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='Lazebnik@unt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'}