Statistical Deferred Nörlund Summability and Korovkin-Type Approximation Theorem

Hari Mohan Srivastava,Bidu Bhusan Jena,Susanta Kumar Paikray

doi:10.3390/math8040636

Hari Mohan Srivastava, Bidu Bhusan Jena + Show 1 more

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https://doi.org/10.3390/math8040636

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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.

Highlights

Introduction and MotivationStatistical convergence plays a vital role as an extension of the classical convergence in the study of convergence analysis of sequence spaces
We prove a new Korovkin-type approximation theorem with algebraic test functions for a real sequence over a Banach space via our proposed methods and demonstrate that our outcome is a non-trivial generalization of ordinary and statistical versions of some well-studied earlier results
We present the following example for the sequence of positive linear operators that does not satisfy the associated conditions of the Korovkin approximation theorems proved previously in [24,33], but it satisfies the conditions of our Theorem 2

Summary

Introduction and Motivation

Statistical convergence plays a vital role as an extension of the classical convergence in the study of convergence analysis of sequence spaces. The credit goes to Fast [1] and Steinhaus [2] for they have independently defined this notion; Zygmund [3] was the first to introduce this idea in the form of “almost convergence” This concept is found in random graph theory (see [4,5]) in the sense that almost convergence, which is same as the statistical convergence, and it means convergence with a probability of 1, whereas in usual statistical convergence the probability is not necessarily 1. Nörlund statistical convergence and used these notions to prove certain Korovkin-type approximation theorem with some new settings. Paikray et al [27] studied a new Korovkin-type theorem involving ( p, q)-integers for statistically deferred Cesàro summability mean. We prove a new Korovkin-type approximation theorem with algebraic test functions for a real sequence over a Banach space via our proposed methods and demonstrate that our outcome is a non-trivial generalization of ordinary and statistical versions of some well-studied earlier results

Preliminaries and Definitions

A New Korovkin-Type Approximation Theorem

Concluding Remarks and Observations