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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: 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

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.

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