Approximation Process Based on Parametric Generalization of Schurer–Kantorovich Operators and their Bivariate Form

Abstract

We construct the Schurer–Kantorovich operators depending on the shape parameter $\alpha \in [0,1]$ which we called $\alpha$-Schurer–Kantorovich operators, and estimate their moments and central moments. We discuss the uniform convergence as well as the rate of convergence in terms of modulus of smoothness and Lipschitz-type functions, and other related results for our new aforementioned operators. Further, we construct the bivariate $\alpha$-Schurer–Kantorovich operators and investigate the degree of convergence with the help of Lipschitz class for bivariate function. Moreover, we discuss the approximation behaviors of bivariate $\alpha$-Schurer–Kantorovich operators for functions having continuous partial derivatives. Statement: We constructed the $\alpha$-Schurer–Kantorovich operators and established several approximation results. Our operators coincide with $\alpha$-Bernstein–Kantorovich operators (for $\nu =0$), Schurer–Kantorovich operators (for $\alpha =1$), and Bernstein–Kantorovich operators (for $\alpha =1$ and $\nu =0$) which means that our operator is stronger than existing in the literature. Thus, we believe that the new operator will open new vistas in this field.