Statistical Approximation for Periodic Functions of Two Variables

S. A. Mohiuddine,Abdullah Alotaibi,M. Mursaleen

doi:10.1155/2013/491768

Abstract

We prove a Korovkin type approximation theorem for a function of two variables by using the notion of statistical summability(C,1,1). We also study the rate of statistical summability(C,1,1)of positive linear operators. Finally we construct an example to show that our result is stronger than those previously proved for Pringsheim's convergence and statistical convergence.

Highlights

Introduction and PreliminariesIn 1951, Fast [1] and Steinhaus [2] independently introduced an extension of the usual concept of sequential limit which is called statistical convergence.The number sequence x is said to be statistically convergent to the number L provided that for each ε > 0, linm 1 n 󵄨󵄨󵄨󵄨{k ≤ n; 󵄨󵄨󵄨󵄨xk − l󵄨󵄨󵄨󵄨 ≥ ε}󵄨󵄨󵄨󵄨 =
We prove a Korovkin type approximation theorem for a function of two variables by using the notion of statistical summability (C, 1, 1)
We study the rate of statistical summability (C, 1, 1) of positive linear operators

Summary

Statistical Approximation for Periodic Functions of Two Variables

We prove a Korovkin type approximation theorem for a function of two variables by using the notion of statistical summability (C, 1, 1). We study the rate of statistical summability (C, 1, 1) of positive linear operators. We construct an example to show that our result is stronger than those previously proved for Pringsheim’s convergence and statistical convergence

Introduction and Preliminaries

It is well known that

We obtain