Tauberian Theorems for the Product of Weighted and Cesàro Summability Methods for Double Sequences

Gökşen Fındık,İbrahim Çanak

doi:10.5269/bspm.v38i7.44135

Abstract

In this paper, we obtain necessary and sufficient conditions, under which convergence of a double sequence in Pringsheim's sense follows from its weighted-Cesaro summability. These Tauberian conditions are one-sided or two-sided if it is a sequence of real or complex numbers, respectively.

Highlights

If a sequence is convergent in Pringsheim’s sense, it is summable N, p, q; 1, 1 (C, α, β), N, p, q; 1, 0 (C, α, β) and N, p, q; 0, 1 (C, α, β) to the same number under boundedness condition
That the converse of this statement holds true is possible under some suitable condition which is so-called a Tauberian condition
Any theorem which states that convergence of a double sequence follows from its weighted-Cesaro summability and some Tauberian condition is said to be a Tauberian theorem for the weighted-Cesaro summability

Summary

Introduction

If a sequence (umn) is convergent in Pringsheim’s sense, it is summable N , p, q; 1, 1 (C, α, β), N , p, q; 1, 0 (C, α, β) and N , p, q; 0, 1 (C, α, β) to the same number under boundedness condition. A sequence (umn) is said to be N , p, q; 1, 0 (C, α, β) summable to s if lim m,n→∞

Results

Conclusion