THE NOVEL TAUBERIAN CONDITIONS ASSOCIATED WITH THE (N,PQ) SUMMABILITY OF DOUBLE SEQUENCES

Önder Şentürk Z., SAVAŞ E., ÇANAK İ.

Journal of Nonlinear and Convex Analysis, vol.25, no.9, pp.2317-2335, 2024 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 25 Issue: 9
  • Publication Date: 2024
  • Journal Name: Journal of Nonlinear and Convex Analysis
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Page Numbers: pp.2317-2335
  • Keywords: (N, p, q) summability, convergence in Pringsheim's sense, Double sequences, regularly varying sequences, slowly decreasing sequences, slowly oscillating sequences, Tauberian conditions and theorems, weighted mean summability method
  • Uşak University Affiliated: Yes

Abstract

In this paper, our primary objective is to provide a fresh perspective on the relationship between the (N,pq) 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,pq) 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,pq) 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.