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Removability, capacity and approximation

  • Chapter
Complex Potential Theory

Part of the book series: NATO ASI Series ((ASIC,volume 439))

Abstract

In this paper we are primarily interested in problems of qualitative approximation by holomorphic functions of one complex variable belonging to some fixed class, that is defined by restricting the growth of the functions (L p, 1 < p ≤ ∞) or by requiring certain smoothness (Lip s or C m). Part of the approximation problem consists in understanding the removable sets for the class under consideration and its associated capacity.

In Chapter 1 we deal with bounded analytic functions. We are thus led to the Painlevé problem and analytic capacity. We discuss the solution of the Denjoy conjecture via L 2-estimates for the Cauchy integral on Lipschitz graphs. We then show that the same ideas can be applied to describe removable sets for Lipschitz analytic functions, the role of the Cauchy integral being played by the Beurling transform.

Chapter 2 is devoted to Vitushkin’s Theorem on uniform approximation by rational functions; the simplest available proof is described in detail.

In Chapter 3, problems of approximation by analytic functions in Lipschitz and C m classes are considered. Vitushkin’s scheme and the mapping properties of the Beurling transform are combined to obtain satisfactory answers to the main questions.

In Chapter 4 we discuss the relationship between L p-approximation by analytic functions and spectral synthesis for Sobolev spaces.

Chapter 5 is a survey of recent results about approximation by solutions of elliptic equations in classical Banach spaces.

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Author information

Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193, Bellaterra (Barcelona), Spain

    Joan Verdera

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  1. Joan Verdera
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Editors and Affiliations

  1. Département de Mathématiques et de Statistique, and Centre de Recherches Mathématiques, Université de Montréal, Montréal, Québec, Canada

    Paul M. Gauthier

  2. Département de Mathématiques et de Statistique, Faculté des Arts et des Sciences, Université de Montréal, Montréal, Québec, Canada

    Gert Sabidussi

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Verdera, J. (1994). Removability, capacity and approximation. In: Gauthier, P.M., Sabidussi, G. (eds) Complex Potential Theory. NATO ASI Series, vol 439. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0934-5_10

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  • DOI: https://doi.org/10.1007/978-94-011-0934-5_10

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4403-5

  • Online ISBN: 978-94-011-0934-5

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