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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: $H^\infty$ and $H^2$ denote the Hardy spaces on the open unit disc $D$. Let $\phi$ be a function in $H^\infty$ and $\Vert\phi\Vert _\infty = 1$. If $\phi$ is an inner function and $\phi(0) = 0$, then $\{\phi^n~;~n = 0,1,2,\cdots \}$ is orthogonal in $H^2$. 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 $\phi$ such that $\phi$ is not an inner function and $\{\phi^n\}$ is orthogonal in $H^2$. In this paper, the following is shown: $\{\phi^n\}$ is orthogonal in $H^2$ if and only if there exists a unique probability measure $\nu_0$ on [0,1] with $1 \in $ supp $\nu_0$ such that $N_\phi(z) = {\int^1_{\vert z\vert}} \log \frac{r}{\vert z\vert} d\nu_0(r)$ for nearly all $z$ in $D$ where $N_\phi$ is the Nevanlinna counting function of $\phi$. If $\phi$ is an inner function, then $\nu_0$ is a Dirac measure at $r = 1$.

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Additional Information

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

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