Abstract
The aim of the present work is to introduce notions of statistical convergence, strongly p-Cesàro summability and the statistically Cauchy sequence of order α in paranormed spaces. Some certain topological properties of these new concepts are examined. Furthermore, we introduce the some inclusion relations among them.
Highlights
Zygmund introduced the idea of statistical convergence in [1]
Fast and Steinhaus independently in the same year introduced statistical convergence to assign a limit to sequences that are not convergent in the usual sense
The set of all statistically convergent sequences is denoted by S, and a sequence that is statistically convergent to ξ is denoted by S − lim xk = ξ
Summary
Zygmund introduced the idea of statistical convergence in [1]. Fast and Steinhaus independently in the same year introduced statistical convergence to assign a limit to sequences that are not convergent in the usual sense (see [2,3]).The notion of the asymptotic (or natural) density of a set A ⊂ N is defined such that: δ ( A) = lim n→∞|{k ≤ n : k ∈ A}| , n whenever the limit exists. |{.}| indicates the cardinality of the enclosed set. Zygmund introduced the idea of statistical convergence in [1]. Fast and Steinhaus independently in the same year introduced statistical convergence to assign a limit to sequences that are not convergent in the usual sense (see [2,3]). The notion of the asymptotic (or natural) density of a set A ⊂ N is defined such that: δ ( A) = lim n→∞. A sequence ( xk ) of numbers is called statistically convergent to a number ξ provided that for every ε > 0, lim n→∞. This notion has been used as an effective tool to resolve many problems in ergodic theory, fuzzy set theory, trigonometric series and Banach spaces in the past few years. Many researchers studied related topics with summability theory. (see [4,5]).
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
