Analogues of some fundamental theorems of summability theory - Richard F. Patterson (International Journal of Mathematics and Mathematical Sciences)
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.