A note on Riesz and Nörlund means

Abstract

In a previous paper [3], the author has considered the necessary and sufficient conditions in order that (R, λn, 1)→(N, pn) and |R, λn, 1|→|N, pn|. In the present note in §2, necessary and sufficient conditions in order that (N, pn)→(R, λn, 1) and |N, pn|→|R, λn, 1|, have been discussed. Limitation theorems and the problem of ineffectiveness for ordinary and absolute summability by Norlund means generated by a class of sequences {P n}, figure in § 3, and discussions on “unified theorems of consistency”, for Riesz summability form the subject matter of the last section (§ 4) of this note.