On the completeness and other properties of some function systems in \(L_p\), \(0<p<\infty\)
type: Item
instance of: scholarly article
title: On the completeness and other properties of some function systems in \(L_p\), \(0<p<\infty\)
zbMATH Open document ID: 0913.46029
author: Vadim I. Filippov
published in: Journal of Approximation Theory
publication date: 1999-06-07T00:00:00Z
review text: Let \(\{\varphi_n\}^\infty_{n= 1}\) be a system of functions in \(L_1(a,b)\) (resp. in \(L_2(a,b)\cap L_p(a,b)\), \(0< p<1\)), where \(-\infty\leq a< b\leq+\infty\). Let \(\{\varphi_{n_k}\}\) be a subsystem of \(\{\varphi_n\}^\infty_{n= 1}\) satisfying suitable conditions. It is proved that for any positive integer \(N\), \(\{\varphi_{n_k}\}^\infty_{k=N}\) is a representation system in \(L_1(a,b)\) (resp. in \(L_p(a,b)\)), i.e. for every \(f\in L_1(a,b)\) (resp. for every \(f\in L_p(a,b)\)) there exists a series \(\sum^\infty_{k=N} c_k\varphi_{n_k}\) such that \(\lim_{m\to\infty} \left\| f-\sum^m_{k=N} c_k\varphi_{n_k}\right\|= 0\) (the norm in \(L_1(a,b)\) resp. in \(L_p(a,b)\)).
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 42A65
Mathematics Subject Classification ID: 46B15
zbMATH DE Number: 1214031
zbMATH Keywords: completeness
zbMATH Keywords: representation system
reviewed by: Julian Musielak
MaRDI profile type: Publication
full work available at URL: jath.1998.3176
OpenAlex ID: W2067447727
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DOI: 10.1006/JATH.1998.3176
description: scientific article; zbMATH DE number 1214031