Better numerical approximation by $\lambda$-Durrmeyer-Bernstein type operators

Abstract

The main object of this paper is to construct a new Durrmeyer variant of the $\lambda$-Bernstein type operators which have better features than the classical one. Some results concerning the rate of convergence in terms of the first and second moduli of continuity and asymptotic formulas of these operators are given. Moreover, we define a bivariate case of these operators and investigate the approximation degree by means of the total and partial modulus of continuity and the Peetre's K-functional. A Voronovskaja type asymptotic and Gruss-Voronovskaja theorem for the bivariate operators is also proven. Further, we introduce the associated GBS (Generalized Boolean Sum) operators and determine the order of convergence with the aid of the mixed modulus of smoothness for the Bogel continuous and Bogel differentiable functions. Finally the theoretical results are analyzed by numerical examples.