THE NOVEL TAUBERIAN CONDITIONS ASSOCIATED WITH THE (NPQ) SUMMABILITY OF DOUBLE SEQUENCES

Abstract

In this paper our primary objective is to provide a fresh perspective on the relationship between the (N<inf>p</inf><inf>q</inf>) method which is a product of relevant one-dimensional summability methods and P-convergence for double sequences. To accomplish this objective we establish certain Tauberian conditions that control the behavior of a double sequence in terms of both Oz-oscillation and O-oscillation in certain senses building a bridge between (N<inf>p</inf><inf>q</inf>) summability and P-convergence while imposing certain restrictions on the weight sequences. As special circumstances of our findings we demonstrate that Landau-type Ol condition with respect to (Pm) and (Qn) as well as Hardy-type O condition with respect to (Pm) and (Qn) serve as Tauberian conditions for (N<inf>p</inf><inf>q</inf>) summability under particular additional conditions. Consequently these results encompass all classical Tauberian theorems including conditions such as slow decrease or slow oscillation in certain senses. © 2024 Elsevier B.V. All rights reserved.