Duquesne University

Department of Mathematics and Computer Science

Technical Report #97-01, October 1997

In 1943 R.C. Buck characterized convergent sequences by proving that $x$ is convergent if and only if there exists a regular matrix $A$ which sums every subsequence of $x$. R.P. Agnew generalized R.C. Buck's theorem as follows: if $x$ is a bounded complex sequence and $A$ is a regular matrix, then there exists a subsequence $y$ of $x$ such that every limit point of $x$ is a limit point of $Ay$. In 1970 I.J. Maddox presented the following theorem: if $A$ is a summability matrix for which there is a divergent sequnce $x$ such that $Ay$ is convergent for every subsequence $y$ of $x$, the $A$ sums every bounded sequence. In this paper definitions for "subsequences of a double sequence" and "Pringsheim limit points" of a double sequence are presented. In addition, multidimensional analogues of Agnew, Buck, and Maddox theorems are presented.

Revised .