Thomson’s theorem on mean square polynomial approximation

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3 Scopus citations

Abstract

In 1991, J. E. Thomson determined completely the structure of H2(μ), the closed subspace of L2(μ) that is spanned by the polynomials, whenever μ is a compactly supported measure in the complex plane. As a consequence he was able to show that if H2(μ) ≠ L2(μ), then every function f ∈ H2(μ) admits an analytic extension to a fixed open set Ω, thereby confirming in this context a phenomenon noted earlier in various situations by S. N. Bernštein, S. N. Mergelyan, and others. Here we present a new proof of Thomson’s results, based on Tolsa’s recent work on the semiadditivity of analytic capacity, which gives more information and is applicable to other problems as well.

Original languageEnglish
Pages (from-to)217-238
Number of pages22
JournalSt. Petersburg Mathematical Journal
Volume17
Issue number2
DOIs
StatePublished - 2006

Keywords

  • Analytic capacity
  • Polynomial approximation
  • Subnormal operators

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

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