On weighted statistical convergence based on (p, q)-integers and related approximation theorems for functions of two variables

In this work, we first define a difference operator Delta([m])(p,q) of natural order m with respect to (p, q)-integers. We then introduce the concepts of Lambda([m])(p,q)-statistical convergence, statistical Lambda([m])(p,q)-summability and strong Lambda([m])(p,q)-summability of order gamma by the weighted method. Furthermore, based on the definition of statistical Lambda([m])(p,q)-summability, we prove a Korovkin type approximation theorem for functions of two variables. By using (p, q)-analogue of Bernstein operator of two variables we give an example which shows that proposed method successfully works. Finally, some Voronovskaja type approximation results are obtained. (C) 2016 Elsevier Inc. All rights reserved.