DERIVATIVES OF BERNSTEIN OPERATORS AND SMOOTHNESS WITH JACOBI WEIGHTS

Abstract

Using the modulus of smoothness with {\it Jacobi} weights $ \omega _{\varphi ^\lambda }^2 (f,t)_\omega $, the relationship between the derivatives {\it Bernstein} operators and the smoothness of the function its approximated in the weighted approximation is characterized, an equivalent theorem between {\it Bernstein} operators and the modulus of smoothness with {\it Jacobi} weights is established. The corresponding results without weights are generalized. In addition, we obtain the direct theorem in the approximation with {\it Jacobi} weights by {\it Bernstein} operators.