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Trudy Mat. Inst. Steklova, 2021, Volume 312, Pages 294–312 (Mi tm4130)  

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

Weighted Fourier Inequalities and Boundedness of Variation

Sergey Yu. Tikhonovabc

a Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain
b ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain
c Universitat Autònoma de Barcelona, Plaça Cívica, 08193 Bellaterra (Cerdanyola del Vallès), Spain

Abstract: We study the trigonometric series $\sum _{n=1}^\infty \lambda _n \cos nx$ and $\sum _{n=1}^\infty \lambda _n \sin nx$ with $\{\lambda _n\}$ being a sequence of bounded variation. Let $\psi $ denote the sum of such a series. We obtain necessary and sufficient conditions for the validity of the weighted Fourier inequality $(\int _0^\pi |\psi (x)|^q \omega (x) dx)^{1/q} \le C(\sum _{n=1}^\infty u_n(\sum _{k=n}^\infty |\lambda _{k}-\lambda _{k+1}|)^p )^{1/p}$, $0<p\le q<\infty $, in terms of the weight $\omega $ and the weighted sequence $\{u_n\}$. Applications to the series with general monotone coefficients are given.

Funding Agency Grant Number
Ministerio de Ciencia e Innovación de España MTM 2017-87409-P
Generalitat de Catalunya 2017 SGR 358
This research was partially supported by Ministerio de Ciencia, Innovación y Universidades (grant MTM 2017-87409-P) and Generalitat de Catalunya (grant 2017 SGR 358).


DOI: https://doi.org/10.4213/tm4130

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English version:
Proceedings of the Steklov Institute of Mathematics, 2021, 312, 282–300

Bibliographic databases:

UDC: 517.518.4
MSC: Primary 42A10; Secondary 41A17, 40A30, 42A16
Received: April 15, 2020
Revised: September 16, 2020
Accepted: October 9, 2020

Citation: Sergey Yu. Tikhonov, “Weighted Fourier Inequalities and Boundedness of Variation”, Function Spaces, Approximation Theory, and Related Problems of Analysis, Collected papers. In commemoration of the 115th anniversary of Academician Sergei Mikhailovich Nikol'skii, Trudy Mat. Inst. Steklova, 312, Steklov Math. Inst., Moscow, 2021, 294–312; Proc. Steklov Inst. Math., 312 (2021), 282–300

Citation in format AMSBIB
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\by Sergey~Yu.~Tikhonov
\paper Weighted Fourier Inequalities and Boundedness of Variation
\inbook Function Spaces, Approximation Theory, and Related Problems of Analysis
\bookinfo Collected papers. In commemoration of the 115th anniversary of Academician Sergei Mikhailovich Nikol'skii
\serial Trudy Mat. Inst. Steklova
\yr 2021
\vol 312
\pages 294--312
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4130}
\crossref{https://doi.org/10.4213/tm4130}
\elib{https://elibrary.ru/item.asp?id=46026403}
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\jour Proc. Steklov Inst. Math.
\yr 2021
\vol 312
\pages 282--300
\crossref{https://doi.org/10.1134/S0081543821010193}
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    This publication is cited in the following articles:
    1. A. S. Belov, M. I. Dyachenko, S. Yu. Tikhonov, “Funktsii s obobschenno monotonnymi koeffitsientami Fure”, UMN, 76:6(462) (2021), 3–70  mathnet  crossref
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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