Better approximation by a Durrmeyer variant of $ \alpha- $Baskakov operators

Abstract

The aim of this paper is to study some approximation properties of the Durrmeyer variant of \begin{document}$ \alpha $\end{document}-Baskakov operators \begin{document}$ M_{n,\alpha} $\end{document} proposed by Aral and Erbay [3]. We study the error in the approximation by these operators in terms of the Lipschitz type maximal function and the order of approximation for these operators by means of the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja and Gr\begin{document}$ \ddot{u} $\end{document}ss Voronovskaja type theorems are also established. Next, we modify these operators in order to preserve the test functions \begin{document}$ e_0 $\end{document} and \begin{document}$ e_2 $\end{document} and show that the modified operators give a better rate of convergence. Finally, we present some graphs to illustrate the convergence behaviour of the operators \begin{document}$ M_{n,\alpha} $\end{document} and show the comparison of its rate of approximation vis-a-vis the modified operators.