On a problem by Steklov

Authors:
A. Aptekarev, S. Denisov and D. Tulyakov

Journal:
J. Amer. Math. Soc. **29** (2016), 1117-1165

MSC (2010):
Primary 42C05; Secondary 33D45

Published electronically:
December 24, 2015

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Abstract: Given any , we define the Steklov class to be the set of probability measures on the unit circle , such that for Lebesgue almost every . One can define the orthonormal polynomials with respect to . In this paper, we obtain the sharp estimates on the uniform norms as which settles a question asked by Steklov in 1921. As an important intermediate step, we consider the following variational problem. Fix and define . Then, we prove

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Additional Information

**A. Aptekarev**

Affiliation:
Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, 125047 Moscow, Russia

Email:
aptekaa@keldysh.ru

**S. Denisov**

Affiliation:
Mathematics Department, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53706; and Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, 125047 Moscow, Russia

Email:
denissov@wisc.edu

**D. Tulyakov**

Affiliation:
Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, 125047 Moscow, Russia

Email:
dntulyakov@gmail.com

DOI:
https://doi.org/10.1090/jams/853

Received by editor(s):
March 17, 2014

Received by editor(s) in revised form:
November 12, 2014, July 26, 2015, and October 12, 2015

Published electronically:
December 24, 2015

Additional Notes:
The work on section 3, which was added in the second revision, July 26, 2015, was supported by Russian Science Foundation grant RSCF-14-21-00025. The research of the first and the third authors on the rest of the paper was supported by Grants RFBR 13-01-12430 OFIm, RFBR 14-01-00604 and Program DMS RAS. The work of the second author on the rest of the paper was supported by NSF Grants DMS-1067413, DMS-1464479.

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American Mathematical Society