The approximation of bivariate generalized Bernstein-Durrmeyer type GBS operators

Abstract

In the present paper, we introduce the generalized Bernstein-Durrmeyer type operators and obtain some approximation properties of these operators studied in the space of continuous functions of two variables on a compact set. The rate of convergence of these operators are given by using the modulus of continuity. A Voronovskaya type asymptotic theorem are studied and some differential properties of these operators are proved. Further, we introduce Bernstein-Durrmeyer type GBS (Generalized Boolean Sum) operator by means of Bögel continuous functions which is more extensive than the space of continuous functions. We obtain the degree of approximation for these operators by using the mixed modulus of smoothness and mixed K-functional. Finally, we show comparisons by some illustrative graphics in Maple for the convergence of the operators to some functions.