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

Abstract

<p style='text-indent:20px;'>This article deals with shape preserving and local approximation properties of post-quantum Bernstein bases and operators over arbitrary interval <inline-formula><tex-math id="M3">\begin{document}$ [a, b] $\end{document}</tex-math></inline-formula> defined by Khan and Sharma (Iran J Sci Technol Trans Sci (2021)). The properties for <inline-formula><tex-math id="M4">\begin{document}$ (\mathfrak{p}, \mathfrak{q}) $\end{document}</tex-math></inline-formula>-Bernstein bases and Bézier curves over <inline-formula><tex-math id="M5">\begin{document}$ [a, b] $\end{document}</tex-math></inline-formula> have been given. A de Casteljau algorithm has been discussed. Further we obtain the rate of convergence for <inline-formula><tex-math id="M6">\begin{document}$ (\mathfrak{p}, \mathfrak{q}) $\end{document}</tex-math></inline-formula>-Bernstein operators over <inline-formula><tex-math id="M7">\begin{document}$ [a, b] $\end{document}</tex-math></inline-formula> in terms of Lipschitz type space having two parameters and Lipschitz maximal functions.</p>