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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2018, Volume 24, Number 4, Pages 199–207
DOI: https://doi.org/10.21538/0134-4889-2018-24-4-199-207
(Mi timm1586)
 

This article is cited in 2 scientific papers (total in 2 papers)

Bernstein–Szegő Inequality for the Weyl Derivative of Trigonometric Polynomials in $L_0$

A. O. Leont'evaab

a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
b Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
Full-text PDF (204 kB) Citations (2)
References:
Abstract: In the set $\mathscr{T}_n$ of trigonometric polynomials $f_n$ of order $n$ with complex coefficients, we consider Weyl (fractional) derivatives $f_n^{(\alpha)}$ of real nonnegative order $\alpha$. The inequality $\left\|D^\alpha_\theta f_n\right\|_p\le B_n(\alpha,\theta)_p \|f_n\|_p$ for the Weyl–Szegő operator $D^\alpha_\theta f_n(t)=f_n^{(\alpha)}(t)\cos\theta+\tilde{f}_n^{(\alpha)}(t)\sin\theta$ in the set $\mathscr{T}_n$ of trigonometric polynomials is a generalization of Bernstein's inequality. Such inequalities have been studied for 90 years. G. Szegő obtained the exact inequality $\left\|f_n'\cos\theta+\tilde{f}_n'\sin\theta\right\|_\infty \leq n\left\|f_n\right\|_\infty$ in 1928. Later, A. Zygmund (1933) and A.I. Kozko (1998) showed that, for $p\ge 1$ and real $\alpha\ge 1$, the constant $B_n(\alpha,\theta)_p$ equals $n^\alpha$ for all $\theta\in\mathbb{R}$. The case $p=0$ is of additional interest because it is in this case that $B_n(\alpha,\theta)_p$ is largest over $p\in[0,\infty]$. In 1994 V. V. Arestov showed that, for $\theta=\pi/2$ (in the case of the conjugate polynomial) and integer nonnegative $\alpha$, the quantity $B_n(\alpha,\pi/2)_0$ grows exponentially in $n$ as $4^{n+o(n)}$. It follows from his result that the behavior of the constant for $\theta\neq 2\pi k$ is the same. However, in the case $\theta=2\pi k$ and $\alpha\in\mathbb{N}$, Arestov showed in 1979 that the exact constant is $n^\alpha$. The author investigated Bernstein's inequality in the case $p=0$ for positive noninteger $\alpha$ earlier (2018). The logarithmic asymptotics of the exact constant was obtained: $\sqrt[n]{B_n(\alpha,0)_0}\to 4$ as $n\to\infty$. In the present paper, this result is generalized to all $\theta \in \mathbb{R}$.
Keywords: trigonometric polynomial, Weyl derivative, conjugate polynomial, Bernstein–Szeg\H{o
Funding agency Grant number
Russian Foundation for Basic Research 18-01-00336
Ministry of Education and Science of the Russian Federation 02.A03.21.0006
This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).
Received: 01.07.2018
Revised: 01.10.2018
Accepted: 15.10.2018
English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2020, Volume 308, Issue 1, Pages S127–S134
DOI: https://doi.org/10.1134/S0081543820020108
Bibliographic databases:
Document Type: Article
UDC: 517.977
MSC: 42A05, 41A17, 26A33
Language: Russian
Citation: A. O. Leont'eva, “Bernstein–Szegő Inequality for the Weyl Derivative of Trigonometric Polynomials in $L_0$”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 4, 2018, 199–207; Proc. Steklov Inst. Math. (Suppl.), 308, suppl. 1 (2020), S127–S134
Citation in format AMSBIB
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\by A.~O.~Leont'eva
\paper Bernstein--Szeg\H{o} Inequality for the Weyl Derivative of Trigonometric Polynomials~in~$L_0$
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2018
\vol 24
\issue 4
\pages 199--207
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\crossref{https://doi.org/10.21538/0134-4889-2018-24-4-199-207}
\elib{https://elibrary.ru/item.asp?id=36517710}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2020
\vol 308
\issue , suppl. 1
\pages S127--S134
\crossref{https://doi.org/10.1134/S0081543820020108}
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