Better degree of approximation by modified Bernstein-Durrmeyer type operators

Abstract

<p style='text-indent:20px;'>In the present article we investigate a Durrmeyer variant of the generalized Bernstein-operators based on a function <inline-formula><tex-math id="M1">\begin{document}$ \tau(x), $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M2">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> is infinitely differentiable function on <inline-formula><tex-math id="M3">\begin{document}$ [0, 1], \; \tau(0) = 0, \tau(1) = 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \tau^{\prime }(x)&gt;0, \;\forall\;\; x\in[0, 1]. $\end{document}</tex-math></inline-formula> We study the degree of approximation by means of the modulus of continuity and the Ditzian-Totik modulus of smoothness. A Voronovskaja type asymptotic theorem and the approximation of functions with derivatives of bounded variation are also studied. By means of a numerical example, finally we illustrate the convergence of these operators to certain functions through graphs and show a careful choice of the function <inline-formula><tex-math id="M5">\begin{document}$ \tau(x) $\end{document}</tex-math></inline-formula> leads to a better approximation than the generalized Bernstein-Durrmeyer type operators considered by Kajla and Acar [<xref ref-type="bibr" rid="b11">11</xref>].</p>