The analytic capacity of sets in problems of approximation theory

Abstract

CONTENTS Introduction Chapter 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 set Chapter 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 points Chapter 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 § 2 Chapter 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 approximation Chapter 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 points References