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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The measure induced by a bounded analytic function on the unit disk may be defined by , where is normalized Lebesgue measure on . 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 , not a constant multiple of an inner function, whose induced measure is rotationally symmetric. This function is not inner, but satisfies if , 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

DOI:
https://doi.org/10.1090/S0894-0347-02-00404-6

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