The Nevanlinna counting functions for Rudin’s orthogonal functions
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- by Takahiko Nakazi PDF
- Proc. Amer. Math. Soc. 131 (2003), 1267-1271 Request permission
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 $\|\phi \|_\infty = 1$. 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 $N_\phi (z) = {\int ^1_{|z|}} \log \frac {r}{|z|} d\nu _0(r)$ 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 $r = 1$.References
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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
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1267-1271
- MSC (2000): Primary 30D50
- DOI: https://doi.org/10.1090/S0002-9939-02-06671-6
- MathSciNet review: 1948119