Approximation of Functions by Generalized Parametric Blending-Type Bernstein Operators

Abstract

In this paper, we introduce a new family of generalized blending-type bivariate Bernstein operators which depends on four parameters $s_{1}$, $s_{2}$, $\alpha _{1}$ and $\alpha _{2}$. Approximation properties of these operators are studied, and we obtain the rate of convergence in terms of mixed and partial modulus of continuities. Moreover, we prove a Korovkin- and a Voronovskaja-type theorems for these operators. The last part of the paper is devoted to the associated GBS operators. In this part, we study degree of approximation of the GBS operators in terms of mixed modulus of continuity. GBS operators obtained here give better approximation than the original operators to the function f(x, y). Finally, approximation properties of the suggested operators and their associated GBS operators are discussed on graphs, for some numerical examples to show how GBS operator gives better approximation to f(x, y). Also, approximation properties of the suggested operators for different values of parameters $s_{1}$, $s_{2}$, $\alpha _{1}$ and $\alpha _{2}$ are illustrated on graphs. It should be mentioned that any increase in $\alpha _{i}$ values or any decrease in $s_{i}$ values gives better approximation of the suggested operators to f(x, y).