Summability factor relations between absolute weighted and Cesàro means

Abstract

By ()U,V, we denote the set of all sequences ϵ=()ϵn such that Σϵnan is summable V whenever Σan is summable U, where U and V are two summability methods. Recently, Sarıgöl has characterized the set ()||C,αk,||trueN‾,pn for k > 1,α > −1 and arbitrary positive sequences ()pn. Now, in the present paper, we characterize the sets ()||C,−1k,||trueN‾,pn, k > 1 and ()||trueN‾,pn,||C,−1k, k ≥ 1 for arbitrary positive sequences ()pn. Hence we extend these results to the range α≥ − 1. In this way, some open problems in this topic are also completed.

A study on absolute summability factors

“…In this context, Sarıgöl [12] has established the result dealing with summability factor of type ∈ (| , | , | , |), for > −1 and > 1 on absolute summability factors , which extends some well-known results of [8][9][10][11]. Also, Hazar Güleç [4] has recently extended these studies to the range ≥ −1 using | , −1| summability method.…”

“…Also, Hazar Güleç [4] has recently extended these studies to the range ≥ −1 using | , −1| summability method.…”

Absolute almost weighted summability methods

“…which reduces to the absolute Cesàro summability | , 1| in Flett [5]' s notation in the special case = 1 for ≥ 0. Many papers concerning the summability | , | was published by several authors (see, [2][3], [5][6][7], [10][11][12][13][14][15]). For example, it is well known that the classical results of Bor [2,3] give sufficient conditions for the equivalance of the summability methods | , 1| and | , | as follows.…”