Generalized Bernstein polynomials

This paper investigates the approximation properties of a generalized Bernstein operator introduced by Cao (J Math Anal Appl 209:140–146, 1997). Employing modulus of continuity and Peetre’s K-functional, we establish asymptotic and quantitative Voronovskaya-type theorems. The non-multiplicativity of the operators is explored through a Grüss–Voronovskaya-type theorem. Additionally, we determine the rate of convergence for functions with derivatives of bounded variation. To validate our theoretical findings, we conduct numerical experiments using MATLAB. These experiments showcase the operators’ convergence behavior and computational efficiency through detailed analysis of tables and graphs. Our results demonstrate the effectiveness of the proposed operators in approximating functions, highlighting their practical applicability.