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

Abstract

Recently, Karsli [15] estimated the convergence rate of the \begin{document}$ q $\end{document}-Bernstein-Durrmeyer operators for functions whose \begin{document}$ q $\end{document}-derivatives are of bounded variation on the interval \begin{document}$ [0, 1] $\end{document}. Inspired by this study, in the present paper we deal with the convergence rate of a \begin{document}$ q $\end{document}- analogue of the Stancu type modified Gamma operators, defined by Karsli et al. [17], for the functions \begin{document}$ \varphi $\end{document} whose \begin{document}$ q $\end{document}-derivatives are of bounded variation on the interval \begin{document}$ [0, \infty ). $\end{document} We present the approximation degree for the operator \begin{document}$ \left( { \mathfrak{S}}_{n, \ell, q}^{(\alpha , \beta )} { \varphi}\right)(\mathfrak{z}) $\end{document} at those points \begin{document}$ \mathfrak{z} $\end{document} at which the one sided q-derivatives\begin{document}$ {D}_{q}^{+}{ \varphi(\mathfrak{z})\; and\; D} _{q}^{-}{ \varphi(\mathfrak{z})} $\end{document} exist.