Generalized Weighted Invariant Mean Based on Fractional Difference Operator with Applications to Approximation Theorems for Functions of Two Variables

In the present article, following a very recent and new approach of Baliarsingh (Alexandria Eng. J., 55(2):1811–1816, 2016), we introduce the notions of statistically $\sigma$-convergence and $\sigma$-statistically convergence by the weighted method with respect to the difference operator ${\Delta }_h^{\alpha ,\beta ,\gamma }$. Some inclusion relations between proposed methods are examined. As an application, we prove a Korovkin type approximation theorem for functions of two variables. Also, by using generalized Meyer–Konig and Zeller operator, we present an example such that our proposed method works but its classical and statistical versions do not work. Finally, we estimate the rate of statistically weighted $\sigma \left({\Delta }_h^{\alpha ,\beta ,\gamma }\right)$-convergence of approximating positive linear operators and give a Voronovskaja-type theorem.