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

 

Rudin's orthogonality problem
and the Nevanlinna counting function


Author: Paul S. Bourdon
Journal: Proc. Amer. Math. Soc. 125 (1997), 1187-1192
MSC (1991): Primary 30D50
MathSciNet review: 1363413
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Abstract: Let $\phi $ be a holomorphic function taking the open unit disk $U$ into itself. We show that the set of nonnegative powers of $\phi $ is orthogonal in $L^2(\partial U)$ if and only if the Nevanlinna counting function of $\phi $, $N_\phi $, is essentially radial. As a corollary, we obtain that the orthogonality of $\{\phi ^n: n=0,1,2,\ldots \}$ for a univalent $\phi $ implies $\phi (z) = \alpha z$ for some constant $\alpha $. We also show that if $\{\phi ^n: n=0,1,2,\ldots \}$ is orthogonal, then the closure of $\phi (U)$ must be a disk.


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Additional Information

Paul S. Bourdon
Affiliation: Department of Mathematics, Washington and Lee University, Lexington, Virginia 24450
Email: pbourdon@wlu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03694-0
PII: S 0002-9939(97)03694-0
Received by editor(s): October 27, 1995
Additional Notes: The author’s research was supported in part by the National Science Foundation (DMS 9401206).
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1997 American Mathematical Society