Korovkin Type Approximation Theorems Proved viaWeighted ??-equistatistical Convergence for Bivariate Functions

Abstract

Statistical convergence was extended to weighted statistical convergence in [24], by using a sequence of real numbers s(k), satisfying some conditions. Later, weighted statistical convergence was considered in [35] and [19] with modified conditions on s(k). Weighted statistical convergence is an extension of statistical convergence in the sense that, for s(k) = 1, for all k, it reduces to statistical convergence. A definition of weighted alpha beta-statistical convergence of order gamma, considered in [25] does not have this property. To remove this extension problem the definition given in [25] 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 s(k) = 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.