The Nevanlinna counting functions for Rudin's orthogonal functions

Author:
Takahiko Nakazi

Journal:
Proc. Amer. Math. Soc. **131** (2003), 1267-1271

MSC (2000):
Primary 30D50

DOI:
https://doi.org/10.1090/S0002-9939-02-06671-6

Published electronically:
September 5, 2002

MathSciNet review:
1948119

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Abstract | References | Similar Articles | Additional Information

Abstract: and denote the Hardy spaces on the open unit disc . Let be a function in and . If is an inner function and , then is orthogonal in . W.Rudin asked if the converse is true and C. Sundberg and C. Bishop showed that the converse is not true. Therefore there exists a function such that is not an inner function and is orthogonal in . In this paper, the following is shown: is orthogonal in if and only if there exists a unique probability measure on [0,1] with supp such that for nearly all in where is the Nevanlinna counting function of . If is an inner function, then is a Dirac measure at .

**1.**C. Bishop, Orthogonal functions in , preprint**2.**P. S. Bourdon, Rudin's orthogonality problem and the Nevanlinna counting function, Proc. Amer. Math. Soc. 125(1997), 1187-1192. MR**98b:30034****3.**J. A. Cima, B. Korenblum and M. Stessin, On Rudin's orthogonality and independence problem, preprint**4.**W. Rudin, Real and Complex Analysis, Third Edition, McGraw-Hill, Inc. MR**88k:00002****5.**W. Rudin, A generalization of a theorem of Frostman, Math. Scand. 21(1967), 136-143. MR**38:3463****6.**J. H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag. MR**94k:47049****7.**A. L. Shields, Weighted shift operators and analytic function theory, Math. Surveys, Amer. Math. Soc. 13, 49-128. MR**50:14341****8.**C. Sundberg, Measures induced by analytic functions and a problem of Walter Rudin, preprint

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Additional Information

**Takahiko Nakazi**

Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

Email:
nakazi@math.sci.hokudai.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-02-06671-6

Received by editor(s):
December 22, 2000

Received by editor(s) in revised form:
December 6, 2001

Published electronically:
September 5, 2002

Additional Notes:
This research was partially supported by Grant-in-Aid for Scientific Research, Ministry of Education

Communicated by:
Juha M. Heinonen

Article copyright:
© Copyright 2002
American Mathematical Society