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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Measures induced by analytic functions and a problem of Walter Rudin
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by Carl Sundberg PDF
J. Amer. Math. Soc. 16 (2003), 69-90 Request permission

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
  • © Copyright 2002 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 16 (2003), 69-90
  • MSC (2000): Primary 30D50
  • DOI: https://doi.org/10.1090/S0894-0347-02-00404-6
  • MathSciNet review: 1937200