International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 4, Pages 775-784
doi:10.1155/S0161171299227755
The Abel-type transformations into ℓ
Department of mathematics, Savannah state university, Savannah 31404, Georgia, USA
Received 24 November 1997; Revised 16 January 1998
Copyright © 1999 Mulatu Lemma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let t be a sequence in (0,1) that converges to 1, and define the Abel-type matrix Aα,t by ank=(k+α k)tnk+1(1−tn)α+1 for α>−1. The matrix Aα,t determines a sequence-to-sequence variant of the Abel-type power series method of summability introduced by Borwein in [1]. The purpose of
this paper is to study these matrices as mappings into ℓ.
Necessary and sufficient conditions for Aα,t to be ℓ-ℓ,G-ℓ, and Gw-ℓ are established. Also, the strength of Aα,t in the ℓ-ℓ setting is investigated.