On Absolute Summability by Riesz and Generalized Cesàro Means. II

Abstract

1. We will use the terminology of part I [9], including the general assumptions of [9, § 1]. In that paper we had proved that |R, λ, κ| = |C, λ, κ| in case that κ is an integer. Now, we turn to non-integral orders κ.As to ordinary summation, the following inclusion relations (in the customary sense; see [9, end of § 1]) for non-integral κ have been established so far. (Since we are comparing Riesz methods of the same type λ and order κ only, (R, λ, κ) is written (R), etc., for the moment.) (R) ⊆ (C) is a result by Borwein and Russell [2]. (C) ⊆ (R) was proved by Jurkat [3] in the case 0 < κ < 1, and, after Borwein [1], it holds in the case 1 < κ < 2 if(1)(2)(i.e. decreases in the wide sense).