On dual summability methods

Abstract

1. Let A be a summability method given by the sequence-to-sequence transformationWe suppose throughout that, for each nconverges; this is a much weaker assumption than the regularity of A. Then we defineWe also suppose throughout that the sequence {sk} is formed by taking the partial sums of the series Σak; that is to say thatLet A' denote the summability method given by the series-to-sequence transformationFollowing Lorent and Zeller (4), (5), we describe A, A' as dual summability methods. We recall that formally,