A generalization of absolute Rieszian summability

Abstract

then Ea, is said to be absolutely summable by Rieszian means of order k and type X, or summable R, X, k| . The case X), n is of particular interest in this paper. Summability R, n, k| has been shown by J. M. Hyslop [3] to be equivalent to absolute Cesaro summability of order k, or summability I C, k| . One of the principal results shown by Obreschkoff was the consistency of the j R, n, k means; that is, he showed that summability |R n, kI implies summability R, n, k' , where k'> k. In this paper we introduce a method of absolute summability based upon the (a, ,B) method of summability defined by Bosanquet and Linfoot [1]. Just as the Bosanquet-Linfoot method generalized Riesz's arithmetic mean (R, n, a), the method given here will generalize absolute Rieszian summability R, n, a I. DEFINITION 2. A series Ea, is said to be absolutely summable (a, ,B), or summable a, f31, where a >0 or ao=0, f3>0, if for each sufficiently large C,