Derivatives of Bernstein operators and smoothness with Jacobi weights

Using the modulus of smoothness with Jacobi weights ω²_φλ (f, t)ω, the relationship between the derivatives Bernstein operators and the smoothness of the function its approximated in the weighted approximation is characterized, an equivalent theorem between Bernstein operators and the modulus of smoothness with Jacobi weights is established. The corresponding results without weights are generalized. In addition, we obtain the direct theorem in the approximation with Jacobi weights by Bernstein operators.