## Measures induced by analytic functions and a problem of Walter Rudin

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- by Carl Sundberg;
- J. Amer. Math. Soc.
**16**(2003), 69-90 - DOI: https://doi.org/10.1090/S0894-0347-02-00404-6
- Published electronically: September 10, 2002
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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.## References

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## Bibliographic 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