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

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 .

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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