Rate of convergence of Stancu type modified $ q $-Gamma operators for functions with derivatives of bounded variation

Abstract

<p style='text-indent:20px;'>Recently, Karsli [<xref ref-type="bibr" rid="b15">15</xref>] estimated the convergence rate of the <inline-formula><tex-math id="M2">\begin{document}$ q $\end{document}</tex-math></inline-formula>-Bernstein-Durrmeyer operators for functions whose <inline-formula><tex-math id="M3">\begin{document}$ q $\end{document}</tex-math></inline-formula>-derivatives are of bounded variation on the interval <inline-formula><tex-math id="M4">\begin{document}$ [0, 1] $\end{document}</tex-math></inline-formula>. Inspired by this study, in the present paper we deal with the convergence rate of a <inline-formula><tex-math id="M5">\begin{document}$ q $\end{document}</tex-math></inline-formula>- analogue of the Stancu type modified Gamma operators, defined by Karsli et al. [<xref ref-type="bibr" rid="b17">17</xref>], for the functions <inline-formula><tex-math id="M6">\begin{document}$ \varphi $\end{document}</tex-math></inline-formula> whose <inline-formula><tex-math id="M7">\begin{document}$ q $\end{document}</tex-math></inline-formula>-derivatives are of bounded variation on the interval <inline-formula><tex-math id="M8">\begin{document}$ [0, \infty ). $\end{document}</tex-math></inline-formula> We present the approximation degree for the operator <inline-formula><tex-math id="M9">\begin{document}$ \left( { \mathfrak{S}}_{n, \ell, q}^{(\alpha , \beta )} { \varphi}\right)(\mathfrak{z}) $\end{document}</tex-math></inline-formula> at those points <inline-formula><tex-math id="M10">\begin{document}$ \mathfrak{z} $\end{document}</tex-math></inline-formula> at which the one sided q-derivatives<inline-formula><tex-math id="M11">\begin{document}$ {D}_{q}^{+}{ \varphi(\mathfrak{z})\; and\; D} _{q}^{-}{ \varphi(\mathfrak{z})} $\end{document}</tex-math></inline-formula> exist.</p>