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It is well known that if a regular matrix sums every subsequence of a sequence x> then x converges. It follows trivially from this result and row finiteness of the Ces&ro summability matrix that if A is a regular matrix such that Ay is Cesέro summable for every subsequence y of x, then x is convergent (not merely Cesaro summable). The purpose of the present paper is to give some general results of this type involving matrix methods that are not necessarily row finite. For example, it is shown that if T is any regular matrix summability method and A is a regular matrix such that Ay is absolutely T-summable for every stretching y of x, then x is absolutely convergent. This is done without assuming that x is bounded, and consequently, without the benefit of associativity.

Summability of matrix transforms of stretchings and subsequences