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Measures induced by analytic functions and a problem of Walter Rudin


Author: Carl Sundberg
Journal: J. Amer. Math. Soc. 16 (2003), 69-90
MSC (2000): Primary 30D50
DOI: https://doi.org/10.1090/S0894-0347-02-00404-6
Published electronically: September 10, 2002
MathSciNet review: 1937200
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Abstract: The measure $\mu _\varphi$ induced by a bounded analytic function $\varphi$ on the unit disk $U$ may be defined by $\mu _\varphi (E)=m(\varphi ^{-1}(E))$, where $m$ is normalized Lebesgue measure on $\partial U$. We discuss the problem of characterizing such measures, and produce some necessary conditions which we conjecture are sufficient. Our main results are a construction showing that our conjectured sufficient conditions are sufficient for a measure to be weakly approximable by induced measures, and a construction of a function $\varphi$, not a constant multiple of an inner function, whose induced measure is rotationally symmetric. This function is not inner, but satisfies $\int \varphi \left (e^{i\theta }\right )^m\overline {\varphi \left (e^{i\theta }\right )^n} \frac {d\theta }{2\pi }=0$ if $m\ne n$, thus answering a question posed by Walter Rudin.


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Additional Information

Carl Sundberg
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300
Email: sundberg@math.utk.edu

Received by editor(s): May 5, 2000
Received by editor(s) in revised form: August 5, 2002
Published electronically: September 10, 2002
Additional Notes: Research supported in part by the National Science Foundation
Article copyright: © Copyright 2002 American Mathematical Society