Modular spaces of sequences connected with some methods of summability
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.
scientific article; zbMATH DE number 1123828