Fundamental theorems of summability theory for a new type of subsequences of double sequences

Abstract

. In 2000, the notion of a subsequence of a double sequence was introduced [3]. Using this de fi nition, a multidimensional analogue to a result from H. Steinhaus, that states that for any regular matrix A there exists a sequence of zeros and ones that is not A -summable, was proved. Additionally, an analogue of a result of R. C. Buck that states that a sequence x is convergent if and only if there exists a regular matrix A that sums every subsequence of x was presented. However, this de fi nition imposes a restrictive condition on the entries of the double sequence that can be considered for the subsequence. In this article, we introduce a less restrictive new de fi nition of a subsequence. We denote them by β -subsequences of a double sequence and show that analogues to these two fundamental theorems of summability still hold for these new subsequences.