International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 1, Pages 1-9

Analogues of some fundamental theorems of summability theory

Richard F. Patterson

Department of Mathematics and Computer Science, Duquesne University, 440 College Hall, Pittsburgh 15282, PA, USA

Received 18 February 1998

Copyright © 2000 Richard F. Patterson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.