Korovkin Second Theorem via -Statistical -Summability

M Mursaleen,A Kiliçman

doi:10.1155/2013/598963

Abstract

Korovkin type approximation theorems are useful tools to check whether a given sequence of positive linear operators on of all continuous functions on the real interval is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, , and in the space as well as for the functions 1, cos, and sin in the space of all continuous 2-periodic functions on the real line. In this paper, we use the notion of -statistical -summability to prove the Korovkin second approximation theorem. We also study the rate of -statistical -summability of a sequence of positive linear operators defined from into .

Highlights

Introduction and PreliminariesLet N be the set of all natural numbers, K ⊆ N, and Kn = {k ≤ n : k ∈ K}
We prove Korovkin second theorem by applying the notion of B-statistical Asummability
We study the rate of B-statistical A-summability of a sequence of positive linear operators defined from C2π(R) into C2π(R)

Summary

Introduction and Preliminaries

A sequence x = (xk) is said to be statistically convergent to L if for every ε > 0, the set Kε := {k ∈ N : |xk − L| ≥ ε} has natural density zero (cf Fast [1]); that is, for each ε > 0, linm In this case, we write L = st − lim x. Note that the first and the second theorems of Korovkin are equivalent to the algebraic and the trigonometric version, respectively, of the classical Weierstrass approximation theorem [19] Such type of approximation theorems has been proved by many authors by using the concept of statistical convergence and its variants, for example, [20,21,22,23,24,25,26,27,28]. We study the rate of B-statistical A-summability of a sequence of positive linear operators defined from C2π(R) into C2π(R)

Main Result

Rate of B-Statistical A-Summability

Example and Concluding Remark