Summability of double sequences by weighted mean methods and Tauberian conditions for convergence in Pringsheim's sense

Ferenc Móricz,U Stadtmüller

doi:10.1155/s0161171204403329

Abstract

After a brief summary of Tauberian conditions for ordinary sequences of numbers, we consider summability of double sequences of real or complex numbers by weighted mean methods which are not necessarily products of related weighted mean methods in one variable. Our goal is to obtain Tauberian conditions under which convergence of a double sequence follows from its summability, where convergence is understood in Pringsheim's sense. In the case of double sequences of real numbers, we present necessary and sufficient Tauberian conditions, which are so-called one-sided conditions. Corollaries allow these Tauberian conditions to be replaced by Schmidt-type slow decrease conditions. For double sequences of complex numbers, we present necessary and sufficient so-called two-sided Tauberian conditions. In particular, these conditions are satisfied if the summable double sequence is slowly oscillating.

Highlights

We begin with a brief and concise summary of the corresponding well-known results for single sequences
The sequence is said to be summable by the weighted mean method determined by the sequence p, in short, summable (N, p), if the sequence converges to a finite limit s; in symbols, sm → s(N, p)
The following special case is called the condition of slow oscillation: inf lim sup max sk − sm = 0

Summary

Introduction

We begin with a brief and concise summary of the corresponding well-known results for single sequences. Let λ = (λ(m)), where λ(m) > m for all m, be an increasing sequence of natural numbers such that lim inf Pλ(m) > 1, m→∞ Pm and denote by Λu the set of all such sequences λ. The following theorem was proved in [8]: for a given sequence (sm) of real numbers, sm → s(N, p) implies sm → s if and only if λ(m) inf λ∈Λu lim sup m→∞

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Results

Conclusion