On Tauberian conditions for weighted generators of triple sequences
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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 $\left(z_{lmn}\right)$ concerning the sequences $\left(P_{l}\right)$, $\left(Q_{m}\right)$, and $\left(R_{n}\right)$, 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 $\left(P_{l}\right)$, $\left(Q_{m}\right)$, and $\left(R_{n}\right)$, 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.
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