On approximation to discrete q-derivatives of functions via q-Bernstein-Schurer operators

In the present paper, we shall investigate the pointwise approximation properties of the \begin{document}$ q- $\end{document} analogue of the Bernstein-Schurer operators and estimate the rate of pointwise convergence of these operators to the functions \begin{document}$ f $\end{document} whose \begin{document}$ q- $\end{document} derivatives are bounded variation on the interval \begin{document}$ [0,1+p]. $\end{document} We give an estimate for the rate of convergence of the operator \begin{document}$ \left( B_{n,p,q}f\right) $\end{document} at those points \begin{document}$ x $\end{document} at which the one sided \begin{document}$ q- $\end{document} derivatives \begin{document}$D_{q}^{+}f(x) $\end{document} and \begin{document}$ D_{q}^{-}f(x) $\end{document} exist. We shall also prove that the operators \begin{document}$ \left( B_{n,p,q}f\right) (x) $\end{document} converge to the limit \begin{document}$ f(x) $\end{document} . As a continuation of the very recent and initial study of the author deals with the pointwise approximation of the \begin{document}$ q- $\end{document} Bernstein Durrmeyer operators [ 12 ] at those points \begin{document}$ x $\end{document} at which the one sided \begin{document}$ q- $\end{document} derivatives \begin{document}$ D_{q}^{+}f(x) $\end{document} and \begin{document}$ D_{q}^{-}f(x) $\end{document} exist, this study provides (or presents) a forward work on the approximation of \begin{document}$ q $\end{document} -analogue of the Schurer type operators in the space of \begin{document}$ D_{q}BV $\end{document} .