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
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.
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.