Better approximation by a Durrmeyer variant of <inline-formula><tex-math id="M1">$ \alpha- $</tex-math></inline-formula>Baskakov operators

Abstract

&lt;p style='text-indent:20px;'&gt;The aim of this paper is to study some approximation properties of the Durrmeyer variant of &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \alpha $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-Baskakov operators &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ M_{n,\alpha} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; proposed by Aral and Erbay [&lt;xref ref-type="bibr" rid="b3"&gt;3&lt;/xref&gt;]. We study the error in the approximation by these operators in terms of the Lipschitz type maximal function and the order of approximation for these operators by means of the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja and Gr&lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ \ddot{u} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;ss Voronovskaja type theorems are also established. Next, we modify these operators in order to preserve the test functions &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ e_0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ e_2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and show that the modified operators give a better rate of convergence. Finally, we present some graphs to illustrate the convergence behaviour of the operators &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ M_{n,\alpha} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and show the comparison of its rate of approximation vis-a-vis the modified operators.&lt;/p&gt;

Approximation by Durrmeyer variant of Cheney-Sharma Chlodovsky operators

“…For the more information about the Durrmeyer operators and their properties, readers should refer the following articles (cf. [5,29,23,25,36,31,9,32,28]).…”