Matrix summability of unbounded sequences

DeFranza, J.; Zeller, K.

doi:10.1090/S0002-9939-1992-1094498-1

Matrix summability of unbounded sequences
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by J. DeFranza and K. Zeller PDF

Proc. Amer. Math. Soc. 115 (1992), 171-175 Request permission

Abstract:

A well-known result of Mazur and Orlicz states that a matrix method strictly stronger than convergence sums not only bounded sequences but unbounded sequences. We consider the question of whether a matrix method strictly stronger than convergence will also sum a sequence with series terms (differences) constituting an unbounded sequence. This is equivalent to the series to sequence convergence domain of the matrix containing an unbounded sequence. A simple criterion is given showing in many cases the answer is positive. Counterexamples of three types are considered; triangles that are not perfect, perfect row finite matrices, and perfect triangles.

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