On Tauberian conditions for weighted generators of triple sequences

Asif Hussain Jan; Tanweer  Jalal

doi:10.11568/kjm.2024.32.4.703

Korean J. Math. Vol. 32 No. 4 (2024) pp.703-725
DOI: https://doi.org/10.11568/kjm.2024.32.4.703

PDF

Published: Dec 30, 2024

Keywords:

Triple sequences,

(\bar{N}, p, q, r)

summability, Slowly decreasing sequences, Weighted generator sequences, Tauberian conditions and theorems, Weighted mean summability method

Mathematics Subject Classification:

40A05, 40E05, 40G99

Asif Hussain Jan

Department of Mathematics, National Institute of Technology, Hazratbal, Srinagar -190006, Jammu and Kashmir, India.

https://orcid.org/0000-0001-9655-227X

Tanweer Jalal

Department of Mathematics, National Institute of Technology, Hazratbal, Srinagar -190006, Jammu and Kashmir, India.

Abstract

This paper introduces a novel perspective on how the $(\bar{N}, p, q, r)$ method relates to $P$ -convergence for triple sequences. Our main objective is to establish Tauberian conditions that govern the behavior of the weighted generator sequence $(z_{l m n})$ concerning the sequences $(P_{l})$ , $(Q_{m})$ , and $(R_{n})$ , aiming to offer a fresh interpretation. These conditions manage the $O_{L}$ - and $O$ -oscillatory properties and establish a link from $(\bar{N}, p, q, r)$ summability to $P$ -convergence, contingent upon specific constraints on the weight sequences. Furthermore, we demonstrate that particular instances, such as the $O_{L}$ -condition of Landau type and the $O$ -condition of Hardy type concerning $(P_{l})$ , $(Q_{m})$ , and $(R_{n})$ , serve as Tauberian conditions for $(\bar{N}, p, q, r)$ summability under additional conditions. Thus, our findings encompass traditional Tauberian theorems, including conditions related to gradual decline and slow oscillation in specific scenarios.

This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License.

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