On universal Fourier series in the Walsh system
type: Item
instance of: scholarly article
title: On universal Fourier series in the Walsh system
zbMATH Open document ID: 1503.42025
DOI: 10.1134/S0037446622050068
published in: Siberian Mathematical Journal
publication date: 2022-10-06T00:00:00Z
review text: The article is dedicated to the study of the existence of universal functions whose Fourier series with respect to the Walsh system are universal in the class of all measurable functions in the sense of signs. For this end, let \(c_k(U)=\int_a^bU(x)\varphi_k(x)dx\) be the Fourier coefficients of the function \(U\in L^1[a,b]\), with respect to an orthonormal system \(\Phi=\{\varphi_k\}_k\) defined on the interval \([a,b]\) and let \(S_m(U,\varphi)(x)=\sum_{k=1}^mc_k(U)\varphi_k(x)\) be the partial sums. Therefore, a function \(U\in L^1[a,b]\) for the space \(S\) with respect to the system \(\Phi=\{\varphi_k\}_k\) is universal in the sense of signs if for each \(f\in S\), we can find a sequence of signs \(\{\delta_k\}_k\) with \(\delta_k=\pm 1\) for \(k\geq 1\), for which the series \(\sum_{k\geq 1} \delta_kc_k(U)\varphi(x)\) converges to \(f(x)\) in \(S\). Then the principal result is the following theorem. Theorem. There exist a sequence \(\{ N_m\}_{m\geq 1}\) of positive integers and \(U\in L^1[0,1]\) whose Fourier-Walsh series has monotone decreasing coefficients and converges in the norm of \(L^1[0,1]\) such that: \par (a) Given \(f\in M[0,1)\), we can find \(\delta_k=\pm 1\) for \(k\geq 1\), such that \[ \lim_{N_m\to \infty}\sum_{k=0}^{N_m}\delta_kc_k(U)W_k(x)=f(x) \] almost everywhere on \([0,1)\) \par (b) \(U\) is universal for \(M[0,1)\) (in the case of convergence in measure) in the Walsh system in the sense of signs. Remark. \par (i) There is a typo on page 868, line 30, first equation; \[ \lim_{k\to \infty}\int_a^b|f_k(x)-f(x)|dx=0\text{ instead of } \lim_{k\to \infty}\int_0^1|f_k(x)-f(x)|dx. \] \par (ii) There is a typo on page 869, line 1, \[ A=\{x\in[a,b), |f(x)|<\infty\}\text{ and }A^{\pm}=\{x\in[a,b), f(x)=\pm\infty\} \] instead of \[ A=\{x\in[0,1), |f(x)|<\infty\}\text{ and }A^{\pm}=\{x\in[0,1), f(x)=\pm\infty\} \]
reviewed by: Iris Athamaica Lopez Palacios
Mathematics Subject Classification ID: 42C10
zbMATH DE Number: 7596648
zbMATH Keywords: universal functions
zbMATH Keywords: Walsh system
zbMATH Keywords: almost everywhere convergence
zbMATH Keywords: convergence in measure
zbMATH Keywords: measurable function space
author: Martin Grigoryan
MaRDI profile type: Publication
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full work available at URL: s0037446622050068
OpenAlex ID: W4300594463
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description: scientific article; zbMATH DE number 7596648