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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A $*$-closed subalgebra of the Smirnov class


Author: Stephan Ramon Garcia
Journal: Proc. Amer. Math. Soc. 133 (2005), 2051-2059
MSC (2000): Primary 30D55
Published electronically: January 14, 2005
MathSciNet review: 2137871
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Abstract | References | Similar Articles | Additional Information

Abstract: We study real Smirnov functions and investigate a certain $*$-closed subalgebra of the Smirnov class $N^+$ containing them. Motivated by a result of Aleksandrov, we provide an explicit representation for the space $H^p \cap \overline{H^p}$. This leads to a natural analog of the Riesz projection on a certain quotient space of $L^p$ for $p \in (0,1)$. We also study a Herglotz-like integral transform for singular measures on the unit circle $\partial \mathbb{D} $.


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Additional Information

Stephan Ramon Garcia
Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106-3080
Email: garcias@math.ucsb.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07735-X
PII: S 0002-9939(05)07735-X
Keywords: Riesz projection, Smirnov class, Herglotz integral, singular measure, Beurling's theorem, backward shift, pseudocontinuation
Received by editor(s): February 3, 2004
Received by editor(s) in revised form: March 5, 2004
Published electronically: January 14, 2005
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.