Thomson’s theorem on mean square polynomial approximation

Brennan, J.

doi:10.1090/S1061-0022-06-00901-0

Thomson’s theorem on mean square polynomial approximation
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by J. E. Brennan
Translated by: the author

St. Petersburg Math. J. 17 (2006), 217-238

DOI: https://doi.org/10.1090/S1061-0022-06-00901-0

Published electronically: February 10, 2006

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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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