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2 results

  • Matrix Transformations Based on Dirichlet Convolution

  • Chikkanna Selvaraj, Suguna Selvaraj
  • Journal:
    Canadian Mathematical Bulletin / Volume 40 / Issue 4 / 01 December 1997
    Published online by Cambridge University Press:
    20 November 2018, pp. 498-508
    Print publication:
    01 December 1997
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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.

  • Abel Transformations Into I1

  • J. A. Fridy
  • Journal:
    Canadian Mathematical Bulletin / Volume 25 / Issue 4 / 01 December 1982
    Published online by Cambridge University Press:
    20 November 2018, pp. 421-427
    Print publication:
    01 December 1982
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  • Let t be a sequence in (0,1) that converges to 0, and define the Abel matrix At by ank = tn(1-tn)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 l1 into l1 if and only if t is in l1. 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 xk has bounded partial sums, then Atx 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.