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Tr. Mat. Inst. Steklova, 2015, Volume 289, Pages 83–106 (Mi tm3628)  

V.A. Steklov's problem of estimating the growth of orthogonal polynomials

A. I. Aptekareva, S. A. Denisovb, D. N. Tulyakova

a Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow, Russia
b Department of Mathematics, University of Wisconsin–Madison, Madison, WI, USA

Abstract: The well-known problem of V.A. Steklov is closely related to the following extremal problem. For a fixed $n\in \mathbb N$, find $M_{n,\delta }=\sup _{\sigma \in S_\delta } \mathopen \|\phi _n\|_{L^\infty (\mathbb T)}$, where $\phi _n(z)$ is an orthonormal polynomial with respect to a measure $\sigma \in S_\delta $ and $S_\delta $ is the Steklov class of probability measures $\sigma $ on the unit circle such that $\sigma '(\theta )\geq \delta /(2\pi )>0$ at every Lebesgue point of $\sigma $. There is an elementary estimate $M_n\lesssim \sqrt n$. E.A. Rakhmanov proved in 1981 that $M_n \gtrsim \sqrt n/ (\ln n)^{3/2}$. Our main result is that $M_n \gtrsim \sqrt n$, i.e., that the elementary estimate is sharp. The paper gives a survey of the results on the solution of this extremal problem and on the general problem of Steklov in the theory of orthogonal polynomials. The paper also analyzes the asymptotics of some trigonometric polynomials defined by Fejér convolutions. These polynomials can be used to construct asymptotic solutions to the extremal problem under consideration.

Funding Agency Grant Number
ОМН РАН 1
Russian Foundation for Basic Research 13-01-12430-ОФИ-м
11-01-00245
National Science Foundation DMS-1067413


DOI: https://doi.org/10.1134/S0371968515020053

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English version:
Proceedings of the Steklov Institute of Mathematics, 2015, 289, 72–95

Bibliographic databases:

UDC: 517.53
Received: January 15, 2014

Citation: A. I. Aptekarev, S. A. Denisov, D. N. Tulyakov, “V.A. Steklov's problem of estimating the growth of orthogonal polynomials”, Selected issues of mathematics and mechanics, Collected papers. In commemoration of the 150th anniversary of Academician Vladimir Andreevich Steklov, Tr. Mat. Inst. Steklova, 289, MAIK Nauka/Interperiodica, Moscow, 2015, 83–106; Proc. Steklov Inst. Math., 289 (2015), 72–95

Citation in format AMSBIB
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\paper V.A.~Steklov's problem of estimating the growth of orthogonal polynomials
\inbook Selected issues of mathematics and mechanics
\bookinfo Collected papers. In commemoration of the 150th anniversary of Academician Vladimir Andreevich Steklov
\serial Tr. Mat. Inst. Steklova
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\vol 289
\pages 83--106
\publ MAIK Nauka/Interperiodica
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  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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