Korovkin type approximation theorems proved via weighted alpha,beta-equistatistical convergence for bivariate functions

Abstract

Statistical convergence was extended to weighted statistical convergence in [21], by using a sequence of real numbers sk, satisfying some conditions. Later, weighted statistical convergence was considered in [31] and [20] with modified conditions on sk. Weighted statistical convergence is an
extension of statistical convergence in the sense that, for sk = 1, for all k, it reduces to statistical convergence. A definition of weighted alpha,beta-statistical convergence of order gamma, considered in [22] does not have this property. To remove this extension problem the definition given in [22] needs some
modifications. In this paper, we introduced the modified version of weighted  alpha, beta-statistical convergence of order
gamma, which is an extension of alpha,beta-statistical convergence of order gamma. Our definition, with
sk = 1, for all k, reduces to alpha,beta-statistical convergence of order gamma. Moreover, we use this definition of weighted alpha,beta-statistical convergence of order gamma, to prove Korovkin type approximation theorems via, weighted alpha,beta-equistatistical convergence of order gamma and weighted
alpha,beta-statistical uniform convergence of order gamma, for bivariate functions on 0, infinity) x [0, infinity). Also we prove Korovkin type approximation theorems via alpha,beta-equistatistical convergence of order gamma  and alpha,beta-statistical uniform convergence of order gamma, for bivariate functions on [0, infinity) x [0, infinity). Some examples of positive linear operators are constructed to show that, our approximation results works, but its classical and statistical cases do not work. Finally, rates of weighted alpha,beta-equistatistical convergence
of order gamma is introduced and discussed.