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Mat. Sb., 2019, Volume 210, Number 7, Pages 94–119 (Mi msb9096)  

Smoothness of functions and Fourier coefficients

M. I. Dyachenkoa, A. B. Mukanovbcd, S. Yu. Tikhonovceb

a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
b Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain
c Centre de Recerca Matemàtica, Bellaterra (Barcelona), Spain
d Kazakhstan Branch of Lomonosov Moscow State University, Astana, Kazakhstan
e Institució Catalana de Recerca i Estudis Avançats, Barcelona, Spain

Abstract: We consider functions represented as trigonometric series with general monotone Fourier coefficients. The main result of the paper is the equivalence of the $L_p$ modulus of smoothness, $1<p<\infty$, of such functions to certain sums of their Fourier coefficients. As applications, for such functions we give a description of the norm in the Besov space and sharp direct and inverse theorems in approximation theory.
Bibliography: 34 titles.

Keywords: Fourier series, general monotone sequences, moduli of smoothness.

Funding Agency Grant Number
Russian Foundation for Basic Research 19-01-00457-a
Ministry of Education and Science of the Republic of Kazakhstan AP05132590
AP05132071
Ministerio de Ciencia e Innovación de España MTM2017-87409-P
Generalitat de Catalunya 2017 SGR 358
M. I. Dyachenko's research was partially supported by the Russian Foundation for Basic Research (grant no. 19-01-00457-a) and the Ministry of Education and Science of the Republic of Kazakhstan (grant no. AP05132590). A. B. Mukanov's research was supported by the Ministry of Education and Science of the Republic of Kazakhstan (grant no. AP05132071). S. Yu. Tikhonov's research was partially supported by Ministerio de Ciencia, Innovación y Universidades (grant no. MTM2017-87409-P) and Generalitat de Catalunya (2017 SGR 358).

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm9096

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English version:
Sbornik: Mathematics, 2019, 210:7, 994–1018

Bibliographic databases:

UDC: 517.518.4+517.518.83
MSC: Primary 42A32; Secondary 26A16, 42A16, 46E35
Received: 08.03.2018 and 06.12.2018

Citation: M. I. Dyachenko, A. B. Mukanov, S. Yu. Tikhonov, “Smoothness of functions and Fourier coefficients”, Mat. Sb., 210:7 (2019), 94–119; Sb. Math., 210:7 (2019), 994–1018

Citation in format AMSBIB
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