Convergence of Derivatives of Generalized Bernstein Operators (Analysis in Theory and Applications, 2012)
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 $n→∞$ 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 $B_∞( f,q;x)$ as $q→ 1^−.$
Keywords: limit $q$-Bernstein operators; derivative of $q$-Bernstein polynomial; convergence; rate.