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

HTML articles powered by AMS MathViewer

- 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
- PDF | 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.## References

- Sheldon Axler, Paul Bourdon, and Wade Ramey,
*Harmonic function theory*, Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 1992. MR**1184139**, DOI 10.1007/b97238 - Lars V. Ahlfors and Leo Sario,
*Riemann surfaces*, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR**0114911**, DOI 10.1515/9781400874538 - Paul S. Bourdon,
*Rudin’s orthogonality problem and the Nevanlinna counting function*, Proc. Amer. Math. Soc.**125**(1997), no. 4, 1187–1192. MR**1363413**, DOI 10.1090/S0002-9939-97-03694-0 - Dietrich Göhde,
*Zum Prinzip der kontraktiven Abbildung*, Math. Nachr.**30**(1965), 251–258 (German). MR**190718**, DOI 10.1002/mana.19650300312 - Lennart Carleson,
*Interpolations by bounded analytic functions and the corona problem*, Ann. of Math. (2)**76**(1962), 547–559. MR**141789**, DOI 10.2307/1970375 - John B. Conway,
*Functions of one complex variable*, 2nd ed., Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1978. MR**503901**, DOI 10.1007/978-1-4612-6313-5 - Peter L. Duren,
*Theory of $H^{p}$ spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655** - Stephen D. Fisher,
*Function theory on planar domains*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1983. A second course in complex analysis; A Wiley-Interscience Publication. MR**694693** - John B. Garnett,
*Bounded analytic functions*, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR**628971** - P. Erdös and T. Grünwald,
*On polynomials with only real roots*, Ann. of Math. (2)**40**(1939), 537–548. MR**7**, DOI 10.2307/1968938 - Oliver Dimon Kellogg,
*Foundations of potential theory*, Die Grundlehren der mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin-New York, 1967. Reprint from the first edition of 1929. MR**0222317**, DOI 10.1007/978-3-642-86748-4
[L]bL Littlewood, J. E., On inequalities in the theory of functions. - Karl Endel Petersen,
*Brownian motion, Hardy spaces and bounded mean oscillation*, London Mathematical Society Lecture Note Series, No. 28, Cambridge University Press, Cambridge-New York-Melbourne, 1977. MR**0651556**, DOI 10.1017/CBO9780511662386 - Sidney C. Port and Charles J. Stone,
*Brownian motion and classical potential theory*, Probability and Mathematical Statistics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**0492329** - Walter Rudin,
*A generalization of a theorem of Frostman*, Math. Scand.**21**(1967), 136–143 (1968). MR**235151**, DOI 10.7146/math.scand.a-10853 - Joel H. Shapiro,
*The essential norm of a composition operator*, Ann. of Math. (2)**125**(1987), no. 2, 375–404. MR**881273**, DOI 10.2307/1971314 - Saunders MacLane and O. F. G. Schilling,
*Infinite number fields with Noether ideal theories*, Amer. J. Math.**61**(1939), 771–782. MR**19**, DOI 10.2307/2371335

*Proc. London Math. Soc.*(2)

**23**(1925), 481–519.

## 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