Better degree of approximation by modified Bernstein-Durrmeyer type operators

Abstract

In the present article we investigate a Durrmeyer variant of the generalized Bernstein-operators based on a function \begin{document}$ \tau(x), $\end{document} where \begin{document}$ \tau $\end{document} is infinitely differentiable function on \begin{document}$ [0, 1], \; \tau(0) = 0, \tau(1) = 1 $\end{document} and \begin{document}$ \tau^{\prime }(x)>0, \;\forall\;\; x\in[0, 1]. $\end{document} We study the degree of approximation by means of the modulus of continuity and the Ditzian-Totik modulus of smoothness. A Voronovskaja type asymptotic theorem and the approximation of functions with derivatives of bounded variation are also studied. By means of a numerical example, finally we illustrate the convergence of these operators to certain functions through graphs and show a careful choice of the function \begin{document}$ \tau(x) $\end{document} leads to a better approximation than the generalized Bernstein-Durrmeyer type operators considered by Kajla and Acar [11].