Approximation and convergence analysis of quantum variant family of ?-Bernstein Kantorovich operators

Abstract

The primary objective of this work is to construct and analyze a novel Kantorovich-Stancu modification of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-Bernstein-Schurer operators (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in [0,1]$$\end{document}) within the framework of quantum calculus. We establish a Bohman-Korovkin-type theorem to ensure the uniform convergence of these newly extended operators. To evaluate the effectiveness of the approximation process, we derive the rate of convergence and establish global approximation results utilizing the Ditzian-Totik modulus of smoothness and Lipschitz-type maximal functions. Furthermore, we explore the approximation behavior across various function classes, highlighting the versatility of these operators in approximating quantum analytic functions. By providing comprehensive uniform convergence criteria and error estimates, this study offers a deeper insight into the performance and applicability of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-Bernstein-Schurer-Kantorovich framework in modern approximation theory.