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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
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

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