Bounded analytic functions
About: This article is published in Duke Mathematical Journal.The article was published on 1947-03-01. It has received 182 citations till now. The article focuses on the topics: Bounded function & Analytic capacity.
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01 Feb 1950
TL;DR: In this article, the authors give an alternative derivation of results on bounded analytic functions recently obtained by Ahlfors [1] and Garabedian [2] and show that the main idea to be used is more in the nature of a lucky guess than of a method.
Abstract: The objective of this paper is to give an alternative derivation of results on bounded analytic functions recently obtained by Ahlfors [1] and Garabedian [2].1 While it is admitted that the main idea to be used is more in the nature of a lucky guess than of a method, it will be found that the gain in brevity and simplicity of the argument is considerable. As a by-product, we shall also obtain a number of hitherto unknown identities between various domain functions. The basic problem treated in the above-mentioned papers is the following generalization of the classical Schwarz lemma: Given a finite schlicht domain D of connectivity n (n> 1) in the complex zplane and a point r in D, to find a function F(z) with the following properties: (a) F(z) belongs to the family B of analytic functions f(z) which are single-valued and regular in D and satisfy there If(z) I _ 1; (b) I F'(r) I > lf'(r) J, where f(z) is any function in B. Evidently, it is sufficient to solve this problem for any domain D' which is conformally equivalent to D. In particular, we may therefore assume, without restricting the generality of what follows, that D is bounded by analytic curves. It was shown by Ahlfors that F(z) yields a (1, n) conformal mapping of D onto the interior of the unit circle and that n-I of the n zeros of F(z) coincide with the zeros of a single-valued function h(z) which is regular in D with the exception of a simple pole at z = and satisfies -'ih(z)dz>O on the boundary r of D; the nth zero of F(z) is located at zx=. It was subsequently noticed by Garabedian that the function h(z) can be written in the form h(z) = F(z)q(z) where q(z) (Z )-2 is regular in D and that the extremal property of F(z) can be deduced in a very elegant manner from the resulting inequality
1,178 citations
TL;DR: In this article, it was shown that the analytic capacity of a compact set of positive measures can be characterized in terms of the curvature of the measures, and the authors deduced that Θ(E) is semiadditive.
Abstract: Let $\gamma(E)$ be the analytic capacity of a compact set $E$ and let $\gamma_+(E)$ be the capacity of $E$ originated by Cauchy transforms of positive measures. In this paper we prove that $\gamma(E)\approx\gamma_+(E)$ with estimates independent of $E$. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that $\gamma$ is semiadditive, which solves a long standing question of Vitushkin.
293 citations
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01 Jan 1972
TL;DR: In this paper, the cauchy transform and Hausdorff measure are used to approximate the approximation of an approximation to a given function in terms of the number of nodes.Analytic capacity
Abstract: Analytic capacity.- The cauchy transform.- Hausdorff measure.- Some examples.- Applications to approximation.
265 citations
TL;DR: In this article, the authors define the analytic capacity of sets and define the connection between the capacity of a set and measures, and give a generalization of the capacity analogue to the theorem on density points.
Abstract: CONTENTSIntroductionChapter I. The analytic capacity of sets § 1. Definition and some properties of analytic capacity § 2. The connection between the capacity of a set and measures § 3. On removable singularities of analytic functions § 4. The analytic C-capacity of sets § 5. Estimates of the coefficients in the Laurent series § 6. The change in the capacity under a conformal transformation of a set Chapter II. The separation of singularities of functions § 1. The construction of a special system of partitions of unity § 2. Integral representations of continuous functions § 3. Separation of singularities § 4. The approximation of functions in parts § 5. Approximation of functions on sets with empty inner boundary § 6. The additivity of capacity for some special partitions of a setChapter III. Estimation of the Cauchy integral § 1. Statement of the result § 2. Estimate of the Cauchy integral § 3. Estimation of the Cauchy integral along a smooth Lyapunov curve § 4. Proof of the principal theorem § 5. Some consequences § 6. A refinement of the Maximum Principle and the capacity analogue to the theorem on density pointsChapter IV. Classification of functions admitting an approximation by rational fractions § 1. Examples of functions that cannot be approximated by rational fractions § 2. A criterion for the approximability of a function § 3. Properties of the second coefficient in the Laurent series § 4. Proof of the principal lemma § 5. Proof of the theorems of § 2Chapter V. The approximation problem for classes of functions § 1. Removal of the poles of approximating functions from the domain of analyticity of the function being approximated § 2. Necessary conditions for the algebras to coincide § 3. A criterion for the equality of the algebras § 4. Geometrical examples § 5. Some problems in the theory of approximationChapter VI. The approximation of functions on nowhere dense sets § 1. The instability of capacity § 2. A capacity criterion for the approximability of functions on nowhere dense sets § 3. Theorems on the approximation of continuous functions in terms of Banach algebras § 4. A capacity characterization of the Mergelyan function and of peak pointsReferences
228 citations
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TL;DR: In this paper, it was shown that the analytic capacity of a compact set of positive measures can be characterized in terms of the curvature of the measures, and the authors deduced that Θ(E) is semiadditive.
Abstract: Let $\gamma(E)$ be the analytic capacity of a compact set $E$ and let $\gamma_+(E)$ be the capacity of $E$ originated by Cauchy transforms of positive measures. In this paper we prove that $\gamma(E)\approx\gamma_+(E)$ with estimates independent of $E$. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that $\gamma$ is semiadditive, which solves a long standing question of Vitushkin.
227 citations