Statistical Deferred Nörlund Summability and Korovkin-Type Approximation Theorem

Abstract

The concept of the deferred Nörlund equi-statistical convergence was introduced and studied by Srivastava et al. [Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM) 112 (2018), 1487–1501]. In the present paper, we have studied the notion of the deferred Nörlund statistical convergence and the statistical deferred Nörlund summability for sequences of real numbers defined over a Banach space. We have also established a theorem presenting a connection between these two interesting notions. Moreover, based upon our proposed methods, we have proved a new Korovkin-type approximation theorem with algebraic test functions for a sequence of real numbers on a Banach space and demonstrated that our theorem effectively extends and improves most of the earlier existing results (in classical and statistical versions). Finally, we have presented an example involving the generalized Meyer–König and Zeller operators of a real sequence demonstrating that our theorem is a stronger approach than its classical and statistical versions.

Schur Lemma and Uniform Convergence of Series through Convergence Methods

“…There is a strong connection between summability methods and approximation theory-for instance, a connection with Korovkin-type approximation theorems [20,21]. It would be very interesting to describe those properties that should exhibit a general a summability method in order to obtain Korovkin-type approximation theorems.…”

Statistical product convergence of martingale sequences and its applications to Korovkin‐type approximation theorems

“…This concept is extremely valuable in Real Analysis, Functional Analysis, Harmonic Analysis, and other related areas. Here, in this connection, we choose to refer the interested readers to the recent works 12,18‐25 …”

Statistical convergence of martingale difference sequence via deferred weighted mean and Korovkin-type theorems

“…Then the deferred weighted mean for the martingale difference sequence [17]. Moreover, recalling another result of Srivastava et al [20] via deferred Nörlund mean D b a (N, p, q) for real sequence given by…”

“…This concept is extremely valuable in Real Analysis, Functional Analysis, Harmonic Analysis, and so on. Here, we like to refer the interested readers to the recent works [4,13,18,20,21] and [26].…”