Convergence of Derivatives of Generalized Bernstein Operators

Abstract

In the present paper, we obtain estimations of convergence rate derivatives of the $q$-Bernstein polynomials $B_n(f,q_n;x)$ approximating to $f'(x)$ as $n\to\infty$ which is a generalization of that relating the classical case $q_n = 1$. On the other hand, we study the convergence properties of derivatives of the limit $q$-Bernstein operators $B_\infty( f,q;x)$ as $q\to 1^−.$