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Abel Transformations Into I 1

Published online by Cambridge University Press:  20 November 2018

J. A. Fridy
J. A. Fridy*
Affiliation:
Kent State University, Kent Ohio
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Abstract

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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 .

Keywords

40D0540D2540E0540G10

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 4 , 01 December 1982 , pp. 421 - 427
Copyright
Copyright © Canadian Mathematical Society 1982

References

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