Uniformly Summable Double Sequences

Abstract

In 1945 Brudno presented the following important theorem: If A and B are regular summability matrix methods such that every bounded sequence summed by A is also summed by B, then it is summed by B to the same value. In 1960 Petersen extended Brudno's theorem by using uniformly summable methods. The goal of this paper is to extend Petersen's theorem to double sequences by using four dimensional matrix transformations and notion of uniformly summable methods for double sequences. In addition to this extension we shall also present an accessible analogue of this theorem.