Rate of Convergence of \(\lambda\)-Bernstein-Beta type operators

Abstract

We propose a Beta-type integral generalization of the Bernstein operators associated with Bézier bases \(\tilde{p}_{m,l}(\lambda ;x)\) and a shape parameter \(\lambda\). First, we study a Korovkin-type result for the proposed operators and then establish their rate of convergence with the help of the modulus of continuity and Peetre’s K-functional. We present a quantitative Voronovskaja type and a Grüss-Voronovskaja type results to study their rate of convergence. We estimate the error for absolutely continuous functions having derivatives of bounded variation. Finally, we provide some graphical examples to illustrate our theoretical results. Relevance of the work:- The generalized operator (1.5) is a powerful tool that can be used to approximate continuous functions, integrable functions, Lipschitz-type functions, and functions with derivatives of bounded variation on the bounded interval [0, 1]. This operator can also be used to solve differential equations and integral equations.