Geodesic
Measures induced by analytic functions and a problem of Walter Rudin
Sundberg, Carl1
1 Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300
Journal of the American Mathematical Society, Tome 16 (2003) no. 1, pp. 69-90

Voir la notice de l'article provenant de la source American Mathematical Society

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.
DOI : 10.1090/S0894-0347-02-00404-6

Sundberg, Carl 1

1 Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300 
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Sundberg, Carl. Measures induced by analytic functions and a problem of Walter Rudin. Journal of the American Mathematical Society, Tome 16 (2003) no. 1, pp. 69-90. doi: 10.1090/S0894-0347-02-00404-6

[1] Axler, Sheldon, Bourdon, Paul, Ramey, Wade Harmonic function theory 1992

[2] Ahlfors, Lars V., Sario, Leo Riemann surfaces 1960

[3] Bourdon, Paul S. Rudin’s orthogonality problem and the Nevanlinna counting function Proc. Amer. Math. Soc. 1997 1187 1192

[4] Gã¶Hde, Dietrich Zum Prinzip der kontraktiven Abbildung Math. Nachr. 1965 251 258

[5] Carleson, Lennart Interpolations by bounded analytic functions and the corona problem Ann. of Math. (2) 1962 547 559

[6] Conway, John B. Functions of one complex variable 1978

[7] Duren, Peter L. Theory of 𝐻^{𝑝} spaces 1970

[8] Fisher, Stephen D. Function theory on planar domains 1983

[9] Garnett, John B. Bounded analytic functions 1981

[10] Erdã¶S, P., Grã¼Nwald, T. On polynomials with only real roots Ann. of Math. (2) 1939 537 548

[11] Kellogg, Oliver Dimon Foundations of potential theory 1967

[12] Petersen, Karl Endel Brownian motion, Hardy spaces and bounded mean oscillation 1977

[13] Port, Sidney C., Stone, Charles J. Brownian motion and classical potential theory 1978

[14] Rudin, Walter A generalization of a theorem of Frostman Math. Scand. 1967

[15] Shapiro, Joel H. The essential norm of a composition operator Ann. of Math. (2) 1987 375 404

[16] Maclane, Saunders, Schilling, O. F. G. Infinite number fields with Noether ideal theories Amer. J. Math. 1939 771 782

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