International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 1, Pages 1-9
doi:10.1155/S0161171200001782

Analogues of some fundamental theorems of summability theory

Abstract

In 1911, Steinhaus presented the following theorem: if A is a regular matrix then there exists a sequence of 0's and 1's which is not A-summable. In 1943, R. C. Buck characterized convergent sequences as follows: a sequence x is convergent if and only if there exists a regular matrix A which sums every subsequence of x. 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.