On the weighted generator of double sequences and its Tauberian conditions

Önder Z., SAVAŞ E., ÇANAK İ.

Advances in Operator Theory, vol.8, no.3, 2023 (ESCI) identifier

  • Publication Type: Article / Article
  • Volume: 8 Issue: 3
  • Publication Date: 2023
  • Doi Number: 10.1007/s43036-023-00262-0
  • Journal Name: Advances in Operator Theory
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus
  • Keywords: (N¯, p, q) summability, Double sequences, Slowly decreasing sequences, Slowly oscillating sequences, Tauberian conditions and theorems, Weighted generator sequences, Weighted mean summability method
  • Uşak University Affiliated: Yes

Abstract

In this paper, our aim is to make a novel interpretation of the relation between (N¯ , p, q) method and P-convergence for double sequences. In accordance with this aim, we derive some Tauberian conditions, controlling OL - and O-oscillatory behavior of the weighted generator sequence of (umn) in the sense (1, 1) relative to (Pm) and/or (Qn) , from (N¯ , p, q) summability to P-convergence with some restrictions on the weight sequences. As special cases, we indicate that OL -condition of Landau type with respect to (Pm) and (Qn) and O-condition of Hardy type with respect to (Pm) and (Qn) are Tauberian conditions for (N¯ , p, q) summability under some additional conditions. Therefore, these results contain all of the classical Tauberian theorems including slow decrease and slow oscillation conditions in certain senses.