Shape preserving properties of $ (\mathfrak{p}, \mathfrak{q}) $ Bernstein Bèzier curves and corresponding results over $ [a, b] $

Abstract

This article deals with shape preserving and local approximation properties of post-quantum Bernstein bases and operators over arbitrary interval \begin{document}$ [a, b] $\end{document} defined by Khan and Sharma (Iran J Sci Technol Trans Sci (2021)). The properties for \begin{document}$ (\mathfrak{p}, \mathfrak{q}) $\end{document}-Bernstein bases and Bézier curves over \begin{document}$ [a, b] $\end{document} have been given. A de Casteljau algorithm has been discussed. Further we obtain the rate of convergence for \begin{document}$ (\mathfrak{p}, \mathfrak{q}) $\end{document}-Bernstein operators over \begin{document}$ [a, b] $\end{document} in terms of Lipschitz type space having two parameters and Lipschitz maximal functions.