Direct and inverse estimates for Bernstein polynomials (Q1265179)

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