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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, Volume 29, Number 4, Pages 130–139
DOI: https://doi.org/10.21538/0134-4889-2023-29-4-130-139
(Mi timm2042)
 

This article is cited in 1 scientific paper (total in 1 paper)

On Constants in the Bernstein–Szegő Inequality for the Weyl Derivative of Order Less Than Unity of Trigonometric Polynomials and Entire Functions of Exponential Type in the Uniform Norm

A. O. Leont'eva

N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
Full-text PDF (223 kB) Citations (1)
References:
Abstract: The Weyl derivative (fractional derivative) $f_n^{(\alpha)}$ of real nonnegative order $\alpha$ is considered on the set $\mathscr{T}_n$ of trigonometric polynomials $f_n$ of order $n$ with complex coefficients. The constant in the Bernstein–Szegő inequality $\|f_n^{(\alpha)}\cos\theta+\tilde{f}_n^{(\alpha)}\sin\theta\|\le B_n(\alpha,\theta)\|f_n\|$ in the uniform norm is studied. This inequality has been well studied for $\alpha\ge 1$: G. T. Sokolov proved in 1935 that it holds with the constant $n^\alpha$ for all $\theta\in\mathbb{R}$. For $0<\alpha<1$, there is much less information about $B_n(\alpha,\theta)$. In this paper, for $0<\alpha<1$ and $\theta\in\mathbb{R}$, we establish the limit relation $\lim_{n\to\infty}B_n(\alpha,\theta)/n^\alpha=\mathcal{B}(\alpha,\theta)$, where $\mathcal{B}(\alpha,\theta)$ is the sharp constant in the similar inequality for entire functions of exponential type at most $1$ that are bounded on the real line. The value $\theta=-\pi\alpha/2$ corresponds to the Riesz derivative, which is an important particular case of the Weyl–Szegő operator. In this case, we derive exact asymptotics for the quantity $B_n(\alpha)=B_n(\alpha,-\pi\alpha/2)$ as $n\to\infty$
Keywords: trigonometric polynomials, entire functions of exponential type, Weyl–Szegő operator, Riesz derivative, Bernstein inequality, uniform norm.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation 075-02-2023-913
This work was performed as a part of the research conducted in the Ural Mathematical Center and supported by the Ministry of Education and Science of the Russian Federation (agreement no. 075-02-2023-913).
Received: 03.07.2023
Revised: 08.08.2023
Accepted: 14.08.2023
English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2023, Volume 323, Issue 1, Pages S146–S154
DOI: https://doi.org/10.1134/S0081543823060123
Bibliographic databases:
Document Type: Article
UDC: 517.518.86
MSC: 26A33, 41A17
Language: Russian
Citation: A. O. Leont'eva, “On Constants in the Bernstein–Szegő Inequality for the Weyl Derivative of Order Less Than Unity of Trigonometric Polynomials and Entire Functions of Exponential Type in the Uniform Norm”, Trudy Inst. Mat. i Mekh. UrO RAN, 29, no. 4, 2023, 130–139; Proc. Steklov Inst. Math. (Suppl.), 323, suppl. 1 (2023), S146–S154
Citation in format AMSBIB
\Bibitem{Leo23}
\by A.~O.~Leont'eva
\paper On Constants in the Bernstein--Szeg\H{o} Inequality for the Weyl Derivative of Order Less Than Unity of Trigonometric Polynomials and Entire Functions of Exponential Type in the Uniform Norm
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2023
\vol 29
\issue 4
\pages 130--139
\mathnet{http://mi.mathnet.ru/timm2042}
\crossref{https://doi.org/10.21538/0134-4889-2023-29-4-130-139}
\elib{https://elibrary.ru/item.asp?id=54950401}
\edn{https://elibrary.ru/lrydxr}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2023
\vol 323
\issue , suppl. 1
\pages S146--S154
\crossref{https://doi.org/10.1134/S0081543823060123}
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