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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.

mathematical foundations of computing

Better approximation by a Durrmeyer variant of &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;$ \alpha- $&lt;/tex-math&gt;&lt;/inline-formula&gt;Baskakov operators