On Hausdorff's methods of summability. II

Abstract

A general theory of ‘strength’ for Hausdorff methods T of summability was given in the first part of this paper, to which we shall refer in the following as I. Those methods T which are regular (that is, at least as strong as ordinary convergence C), are defined by the linear transformsof a given sequence (sk). Here ø(t) (that is, each of its real and imaginary parts) is of bounded variation (b.v.) in the closed interval 〈0, 1〉 [see I, (4)]. The additional condition for consistency (with C) is that ø(t) is continuous at t = 0, and that ø(1) − ø(0) = 1 [I, H. 3]. We write also T ˜ ø(t).