Quantitative Estimates for Generalized Bernstein–Kantorovich Operators

This paper introduces a Kantorovich-type modification of the generalized Bernstein polynomials proposed by Cao (J Math Anal Appl 209:140–146, 1997). We investigate the convergence properties of these operators, establishing necessary and sufficient conditions for uniform convergence and deriving quantitative estimates for the rate of convergence in terms of the Ditzian–Totik unified modulus of smoothness. Furthermore, we establish an inverse approximation theorem that characterizes the smoothness of the function based on the rate of convergence of the operators. We extend our analysis to the $L_p$-norm setting, proving the convergence of the operators for functions in $L_p[0,1],1\le p<\infty$ and providing quantitative estimates for the rate of convergence in the $L_p$-norm, utilizing the integral modulus of smoothness. Numerical simulations demonstrate the effectiveness of the proposed operators. We observe rapid convergence in practical examples, with significant error reduction as the number of basis functions ( $n$) increases. These findings are supported by detailed numerical experiments and visual representations, showcasing the practical applicability of the operators.