Analogues of some fundamental theorems of summability theory

Abstract

In 1911, Steinhaus presented the following theorem: ifAis a regular matrix then there exists a sequence of 0's and 1's which is notA-summable. In 1943, R. C. Buck characterized convergent sequences as follows: a sequencexis convergent if and only if there exists a regular matrixAwhich sums every subsequence ofx. In this paper, definitions for “subsequences of a double sequence” and “Pringsheim limit points” of a double sequence are introduced. In addition, multidimensional analogues of Steinhaus' and Buck's theorems are proved.