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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
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
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
Linking options:
https://www.mathnet.ru/eng/im8373https://doi.org/10.1070/IM8373 https://www.mathnet.ru/eng/im/v80/i6/p65
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Abstract page: | 443 | Russian version PDF: | 84 | English version PDF: | 10 | References: | 62 | First page: | 19 |
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