Trudy Matematicheskogo Instituta imeni V.A. Steklova
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Guidelines for authors License agreement Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Trudy Mat. Inst. Steklova: Year: Volume: Issue: Page: Find

 Trudy 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 ÎÌÍ ÐÀÍ 1 Russian Foundation for Basic Research 13-01-12430-ÎÔÈ-ì11-01-00245 National Science Foundation DMS-1067413

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

Full text: PDF file (303 kB)
References: PDF file   HTML file

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, Trudy Mat. Inst. Steklova, 289, MAIK Nauka/Interperiodica, Moscow, 2015, 83–106; Proc. Steklov Inst. Math., 289 (2015), 72–95

Citation in format AMSBIB
\Bibitem{AptDenTul15} \by A.~I.~Aptekarev, S.~A.~Denisov, D.~N.~Tulyakov \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 Trudy Mat. Inst. Steklova \yr 2015 \vol 289 \pages 83--106 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3628} \crossref{https://doi.org/10.1134/S0371968515020053} \elib{https://elibrary.ru/item.asp?id=23738463} \transl \jour Proc. Steklov Inst. Math. \yr 2015 \vol 289 \pages 72--95 \crossref{https://doi.org/10.1134/S0081543815040057} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000358577300005} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84938891216} 

• http://mi.mathnet.ru/eng/tm3628
• https://doi.org/10.1134/S0371968515020053
• http://mi.mathnet.ru/eng/tm/v289/p83

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. D. S. Lubinsky, “Local asymptotics for orthonormal polynomials on the unit circle via universality”, J. Anal. Math., 141:1 (2020), 285–304
•  Number of views: This page: 228 Full text: 19 References: 20 First page: 1

 Contact us: math-net2021_12 [at] mi-ras ru Terms of Use Registration to the website Logotypes © Steklov Mathematical Institute RAS, 2021