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. Proc. London Math. Soc. (2) 23 (1925), 481–519.
- 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
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