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Let p = ( p m ) be a sequence that satisfies ( p m ) = ( P m − P m − 1 ), where P − 1 = 0 and P m (cid:13)= 0 for all m ∈ N -
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ii) a rapidly varying sequence of index ∞ (see [3] for more details) -
(i) The symbol u mn = O ( 1 ) means that | u mn | ≤ H for some constant H > 0 and each m , n ≥ n 0 -
b) In case p m = 1 /( m + 1 ) and q n = 1 /( n + 1 ), it leads to the harmonic means (or called the logarithmic means) of a double sequence where P m ∼ log m and Q n ∼ log n for all m , n ∈ N
- Publisher
- Springer Journals
- Copyright
- Copyright © Tusi Mathematical Research Group (TMRG) 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
- ISSN
- 2662-2009
- eISSN
- 2538-225X
- DOI
- 10.1007/s43036-023-00262-0
- Publisher site
- See Article on Publisher Site
Abstract
In this paper, our aim is to make a novel interpretation of the relation between (N¯,p,q)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$({\overline{N}},p,q)$$\end{document} method and P-convergence for double sequences. In accordance with this aim, we derive some Tauberian conditions, controlling OL\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O_L$$\end{document}- and O-oscillatory behavior of the weighted generator sequence of (umn)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(u_{mn})$$\end{document} in the sense (1, 1) relative to (Pm)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(P_m)$$\end{document} and/or (Qn),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(Q_n),$$\end{document} from (N¯,p,q)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$({\overline{N}},p,q)$$\end{document} summability to P-convergence with some restrictions on the weight sequences. As special cases, we indicate that OL\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O_L$$\end{document}-condition of Landau type with respect to (Pm)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(P_m)$$\end{document} and (Qn)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(Q_n)$$\end{document} and O-condition of Hardy type with respect to (Pm)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(P_m)$$\end{document} and (Qn)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(Q_n)$$\end{document} are Tauberian conditions for (N¯,p,q)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$({\overline{N}},p,q)$$\end{document} 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.
Journal
Advances in Operator Theory – Springer Journals
Published: Jul 1, 2023
Keywords: Double sequences; (N¯,p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\overline{N}}, p, q)$$\end{document} summability; Slowly decreasing sequences; Slowly oscillating sequences; Weighted generator sequences; Tauberian conditions and theorems; Weighted mean summability method; 40A05; 40E05; 40G99