A new note on generalized absolute matrix summability

Hs Özarslan,T Ari

doi:10.1186/1029-242x-2012-166

Abstract

This paper gives necessary and sufficient conditions in order that a series should be summable , , whenever is summable . Some new results have also been obtained. MSC:40D25, 40F05, 40G99.

Highlights

Let an be a given infinite series with the partial sums
Defines the sequence of the Riesz means of the sequence generated by the sequence of coefficients
Since Tn( ) + Tn( ) k ≤ k Tn( ) k + Tn( ) k to complete the proof of Theorem, it is sufficient to show that nk– Tn(i) k < ∞ for i =

Summary

Introduction

Let an be a given infinite series with the partial sums (sn). Let (pn) be a sequence of positive numbers such that n. Pn = pv → ∞ as (n → ∞), (P–i = p–i = , i ≥ ). Defines the sequence (tn) of the Riesz means of the sequence (sn) generated by the sequence of coefficients (pn) (see [ ]). The series an is said to be summable |R, pn|k, k ≥ , if (see [ ]). Let A = (anv) be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. A defines the sequence-to-sequence transformation, mapping the sequence s = (sn) to As = (An(s)), where n. Nk– ̄ An(s) k < ∞, where An(s) = An(s) – An– (s)

If we take anv

Qv pv