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This paper is a study of summability methods that are based on Dirichlet convolution. If f(n) is a function on positive integers and x is a sequence such that then x is said to be Af-summable to L. The necessary and sufficient condition for the matrix Af to preserve bounded variation of sequences is established. Also, the matrix Af is investigated as ℓ − ℓ and G − G mappings. The strength of the Af -matrix is also discussed.

Let t be a sequence in (0,1) that converges to 0, and define the Abel matrix At by a nk = t n(1-t n )k . The matrix At determines a sequence-to-sequence variant of the classical Abel summability method. The purpose of this paper is to study these transformations as l-l summability methods: e.g., At maps l 1 into l 1 if and only if t is in l 1. The Abel matrices are shown to be stronger l-l methods than the Euler-Knopp means and the Nӧrlund means. Indeed, if t is in l1 and Σ x k has bounded partial sums, then A t x is in l1. Also, the Abel matrix is shown to be translative in an l-l sense, and an l-l Tauberian theorem is proved for At .

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Matrix Transformations Based on Dirichlet Convolution

Abel Transformations Into I 1

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2 results

Matrix Transformations Based on Dirichlet Convolution

Abel Transformations Into I 1