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Thomson's theorem on mean square polynomial approximation


Author: J. E. Brennan
Translated by: the author
Original publication: Algebra i Analiz, tom 17 (2005), nomer 2.
Journal: St. Petersburg Math. J. 17 (2006), 217-238
MSC (2000): Primary 41A10, 30E10, 31A15, 47B20
DOI: https://doi.org/10.1090/S1061-0022-06-00901-0
Published electronically: February 10, 2006
MathSciNet review: 2159582
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Abstract | References | Similar Articles | Additional Information

Abstract: In 1991, J. E. Thomson determined completely the structure of $ H^2(\mu)$, the closed subspace of $ L^2(\mu)$ that is spanned by the polynomials, whenever $ \mu$ is a compactly supported measure in the complex plane. As a consequence he was able to show that if $ H^2(\mu)\ne L^2(\mu)$, then every function $ f\in H^2(\mu)$ admits an analytic extension to a fixed open set $ \Omega$, 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.


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Additional Information

J. E. Brennan
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: brennan@ms.uky.edu

DOI: https://doi.org/10.1090/S1061-0022-06-00901-0
Keywords: Polynomial approximation, analytic capacity, subnormal operators
Received by editor(s): May 24, 2004
Published electronically: February 10, 2006
Article copyright: © Copyright 2006 American Mathematical Society

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