Direct and Inverse Estimates for Bernstein Polynomials

Abstract.

Direct estimates for the Bernstein operator are presented by the Ditzian—Totik modulus of smoothness $\omega_\phi^2(f,\delta)$ , whereby the step-weight φ is a function such that φ ² is concave. The inverse direction will be established for those step-weights φ for which φ ² and $\varphi^2 / \phi^2, \varphi(x)=\sqrt{x(1-x)}$ , are concave functions. This combines the classical estimate (φ=1 ) and the estimate developed by Ditzian and Totik ($\phi=\varphi$ ). In particular, the cases $\phi=\varphi^\lambda$ , λ∈[0,1] , are included.