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

 

Cauchy transforms of characteristic functions and algebras generated by inner functions


Authors: Alec L. Matheson and Michael I. Stessin
Journal: Proc. Amer. Math. Soc. 133 (2005), 3361-3370
MSC (2000): Primary 46J10; Secondary 46J15, 30D50, 30D55
Published electronically: May 9, 2005
MathSciNet review: 2161161
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Abstract: We prove that Cauchy transforms of characteristic functions of subsets of positive measure of the unit circle are equidistributed in the unit disk in the sense that the $L^p$-closure of the polynomial algebra in these Cauchy transforms coincides with the $L^p$-closure of the polynomial algebra in a canonical inner function. As a corollary to this result we find conditions describing when the polynomial algebra in two singular inner functions determined by point masses is dense in the Hardy spaces $H^p$.


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Additional Information

Alec L. Matheson
Affiliation: Department of Mathematics, Lamar University, Beaumont, Texas 77710
Email: matheson@math.lamar.edu

Michael I. Stessin
Affiliation: Department of Mathematics and Statistics, University at Albany, SUNY, Albany, New York 12222
Email: stessin@math.albany.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07913-X
PII: S 0002-9939(05)07913-X
Received by editor(s): May 5, 2004
Received by editor(s) in revised form: June 28, 2004
Published electronically: May 9, 2005
Additional Notes: This work was accomplished while the first author was visiting the University at Albany. He thanks that institution for the hospitality extended during his visit.
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.