Modular spaces of sequences connected with some methods of summability
type: Item
instance of: scholarly article
title: Modular spaces of sequences connected with some methods of summability
zbMATH Open document ID: 0912.46003
author: Aleksander Waszak
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 1999-05-03T00:00:00Z
review text: Let \(A\) be a non-negative infinite matrix which contains no columns consisting of zeros only (and fulfills further certain conditions), and let \(\varphi\) be a continuous non-decreasing function defined for \(u\geq 0\) such that \(\varphi(0)=0\), \(\varphi >0\) for \(u>0\) and \(\varphi(u)\to \infty\) as \(u\to\infty\). \textit{J. Musielak} and \textit{W. Orlicz} [Stud. Math. 22, 127-146 (1962; Zbl 0111.30501)] and \textit{R. Taberski} [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 8, 453-458 (1960; Zbl 0097.27303)] introduced the method of strong \((A,\varphi) \)-summability and that of strong \(| A,\varphi| \)-summability, respectively, however they investigated them only in the particular case as \(A\) is the Cesàro matrix of order one. The author dealt with the general case already in several papers. In this paper, denoting by \(T^*_\varphi\) and \(T^b_\varphi\) the set of all real sequences being \((A,\varphi) \)-summable to zero and \(| A,\varphi| \)-summable to zero, respectively, he gives in each case sufficient conditions on functions \(\varphi\) and \(\psi\) in order that \(T_\varphi^*\subset T_\psi^b\) (Theorem on page 105) and that \(T_\psi^b\subset T_\varphi^*\) (Theorem on page 107) holds.
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46A80
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 40D25
zbMATH DE Number: 1123828
zbMATH Keywords: sequence spaces
zbMATH Keywords: modular spaces
zbMATH Keywords: summability methods
zbMATH Keywords: inclusion theorem
zbMATH Keywords: strong \((A,\varphi)\)-summability
zbMATH Keywords: Cesàro matrix
reviewed by: Johann Boos
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1123828