The novel Tauberian conditions associated with the $(\overline{N},p,q)$ summability of double sequences
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arXiv
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| Format: | Preprint |
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2023
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| author | Önder, Zerrin Savaş, Ekrem Çanak, İbrahim |
| author_facet | Önder, Zerrin Savaş, Ekrem Çanak, İbrahim |
| contents | In this paper, our primary objective is to provide a fresh perspective on the relationship between the $(\overline{N},p,q)$ 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 $O_L$-oscillation and $O$-oscillation in certain senses, building a bridge between $(\overline{N},p,q)$ summability and $P$-convergence, while imposing certain restrictions on the weight sequences. As special circumstances of our findings, we demonstrate that Landau-type $O_L$ condition with respect to $(P_m)$ and $(Q_n),$ as well as Hardy-type $O$ condition with respect to $(P_m)$ and $(Q_n),$ serve as Tauberian conditions for $(\overline{N},p,q)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_00517 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The novel Tauberian conditions associated with the $(\overline{N},p,q)$ summability of double sequences Önder, Zerrin Savaş, Ekrem Çanak, İbrahim General Mathematics 40A05, 40E05, 40G99 In this paper, our primary objective is to provide a fresh perspective on the relationship between the $(\overline{N},p,q)$ 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 $O_L$-oscillation and $O$-oscillation in certain senses, building a bridge between $(\overline{N},p,q)$ summability and $P$-convergence, while imposing certain restrictions on the weight sequences. As special circumstances of our findings, we demonstrate that Landau-type $O_L$ condition with respect to $(P_m)$ and $(Q_n),$ as well as Hardy-type $O$ condition with respect to $(P_m)$ and $(Q_n),$ serve as Tauberian conditions for $(\overline{N},p,q)$ 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. |
| title | The novel Tauberian conditions associated with the $(\overline{N},p,q)$ summability of double sequences |
| topic | General Mathematics 40A05, 40E05, 40G99 |
| url | https://arxiv.org/abs/2307.00517 |