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ON \(A^{\mathcal{I^{K}}}\)–SUMMABILITY

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Titel: ON \(A^{\mathcal{I^{K}}}\)–SUMMABILITY
Autoren: Chiranjib Choudhury, Shyamal Debnath
Quelle: Ural Mathematical Journal, Vol 8, Iss 1 (2022)
Verlagsinformationen: Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics, 2022.
Publikationsjahr: 2022
Bestand: LCC:Mathematics
Schlagwörter: ideal, filter, \(\mathcal{i}\)-convergence, \(\mathcal{i^{k}}\)-convergence, \(a^{\mathcal{i}}\)-summability, \(a^{\mathcal{i^{k}}}\)-summability., Mathematics, QA1-939
Beschreibung: In this paper, we introduce and investigate the concept of \(A^{\mathcal{I^{K}}}\)-summability as an extension of \(A^{\mathcal{I^{*}}}\)-summability which was recently (2021) introduced by O.H.H.~Edely, where \(A=(a_{nk})_{n,k=1}^{\infty}\) is a non-negative regular matrix and \(\mathcal{I}\) and \(\mathcal{K}\) represent two non-trivial admissible ideals in \(\mathbb{N}\). We study some of its fundamental properties as well as a few inclusion relationships with some other known summability methods. We prove that \(A^{\mathcal{K}}\)-summability always implies \(A^{\mathcal{I^{K}}}\)-summability whereas \(A^{\mathcal{I}}\)-summability not necessarily implies \(A^{\mathcal{I^{K}}}\)-summability. Finally, we give a condition namely \(AP(\mathcal{I},\mathcal{K})\) (which is a natural generalization of the condition \(AP\)) under which \(A^{\mathcal{I}}\)-summability implies \(A^{\mathcal{I^{K}}}\)-summability.
Publikationsart: article
Dateibeschreibung: electronic resource
Sprache: English
ISSN: 2414-3952
Relation: https://umjuran.ru/index.php/umj/article/view/410; https://doaj.org/toc/2414-3952
DOI: 10.15826/umj.2022.1.002
Zugangs-URL: https://doaj.org/article/9568441849ed4fbdbf58c79c6d42164e
Dokumentencode: edsdoj.9568441849ed4fbdbf58c79c6d42164e
Datenbank: Directory of Open Access Journals
Beschreibung
Abstract:In this paper, we introduce and investigate the concept of \(A^{\mathcal{I^{K}}}\)-summability as an extension of \(A^{\mathcal{I^{*}}}\)-summability which was recently (2021) introduced by O.H.H.~Edely, where \(A=(a_{nk})_{n,k=1}^{\infty}\) is a non-negative regular matrix and \(\mathcal{I}\) and \(\mathcal{K}\) represent two non-trivial admissible ideals in \(\mathbb{N}\). We study some of its fundamental properties as well as a few inclusion relationships with some other known summability methods. We prove that \(A^{\mathcal{K}}\)-summability always implies \(A^{\mathcal{I^{K}}}\)-summability whereas \(A^{\mathcal{I}}\)-summability not necessarily implies \(A^{\mathcal{I^{K}}}\)-summability. Finally, we give a condition namely \(AP(\mathcal{I},\mathcal{K})\) (which is a natural generalization of the condition \(AP\)) under which \(A^{\mathcal{I}}\)-summability implies \(A^{\mathcal{I^{K}}}\)-summability.
ISSN:24143952
DOI:10.15826/umj.2022.1.002