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Izvestiya: Mathematics, 2016, Volume 80, Issue 6, Pages 1057–1083
DOI: https://doi.org/10.1070/IM8373
(Mi im8373)
 

This article is cited in 13 scientific papers (total in 13 papers)

Universal functions in ‘correction’ problems guaranteeing the convergence of Fourier–Walsh series

M. G. Grigoryana, K. A. Navasardyanb

a Physical and Mathematical Faculty of Yerevan State University
b Yerevan State University, Faculty of Informatics and Applied Mathematics
References:
Abstract: We prove the existence of a function $g(x)\in L^1[0,1]$ with monotone decreasing Fourier–Walsh coefficients $\{c_k(g)\}_{k=0}^\infty\downarrow$ which is universal in $L^p[0,1]$, $p\geqslant1$, in the sense of modification with respect to the signs of the Fourier coefficients for the Walsh system. In other words, for every function $f\in L^p[0;1]$ and every $\varepsilon>0$ one can find a function $\widetilde f\in L^p[0;1]$ such that the measure $|\{x\in[0;1]\colon f(x)=\widetilde f(x)\}|$ is greater than $1-\varepsilon$, the Fourier series of $\widetilde f(x)$ in the Walsh system converges to $\widetilde f(x)$ in the $L^p[0,1]$-norm and $|c_k(\widetilde f)|=c_k(g)$, $k\in\operatorname{Spec}(\widetilde f)$. We also prove that for every $\varepsilon$, $0<\varepsilon<1$, one can find a measurable set $E\subset [0,1]$ of measure $|E|>1-\varepsilon$ and a function $g\in L^1[0;1]$ with $0<c_{k+1}(g)<c_k(g)$, $k=0,1,2,\dots$, such that for every function $f\in L^1[0,1]$ there is a function $\widetilde f\in L^1[0,1]$ with the following properties: $\widetilde f$ coincides with $f$ on $E$, the Fourier–Walsh series of $\widetilde f(x)$ converges to $\widetilde f(x)$ in the norm of $L^1[0,1]$ and the absolute values of all terms in the sequence of the Fourier–Walsh coefficients of the newly obtained function satisfy $|c_k(\widetilde f)|=c_k(g)$, $k=0,1,2,\dots$ .
Keywords: Fourier coefficients, Walsh system, convergence in the $L^1$-norm.
Received: 30.03.2015
Revised: 29.07.2015
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2016, Volume 80, Issue 6, Pages 65–91
DOI: https://doi.org/10.4213/im8373
Bibliographic databases:
UDC: 517.51
MSC: 26D15, 42C10, 42C20
Language: English
Original paper language: Russian
Citation: M. G. Grigoryan, K. A. Navasardyan, “Universal functions in ‘correction’ problems guaranteeing the convergence of Fourier–Walsh series”, Izv. RAN. Ser. Mat., 80:6 (2016), 65–91; Izv. Math., 80:6 (2016), 1057–1083
Citation in format AMSBIB
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  • This publication is cited in the following 13 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    Russian version PDF:84
    English version PDF:10
    References:62
    First page:19
     
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