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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 $ \delta \in (0,1)$, we define the Steklov class $ S_\delta $ to be the set of probability measures $ \sigma $ on the unit circle $ \mathbb{T}$, such that $ \sigma '(\theta )\ge \delta /(2\pi )>0$ for Lebesgue almost every $ \theta \in [0,2\pi )$. One can define the orthonormal polynomials $ \phi _n(z)$ with respect to $ \sigma \in S_\delta $. In this paper, we obtain the sharp estimates on the uniform norms $ \Vert\phi _n\Vert _{L^\infty (\mathbb{T})}$ as $ n\to \infty $ which settles a question asked by Steklov in 1921. As an important intermediate step, we consider the following variational problem. Fix $ n\in \mathbb{N}$ and define $ M_{n,\delta }=\smash {\sup \limits _{\sigma \in S_\delta }}\Vert\phi _n\Vert _{L^\infty (\mathbb{T})}$. Then, we prove

$\displaystyle C(\delta )\sqrt n < M_{n,\delta }\le \sqrt {\frac {n+1}\delta }\,\,. $

A new method is developed that can be used to study other important variational problems. For instance, we prove the sharp estimates for the polynomial entropy in the Steklov class.

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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.
Article copyright: © Copyright 2015 American Mathematical Society