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.