DIRECT AND INVERSE ESTIMATES FOR BERNSTEIN POLYNOMIALS

Direct estimates for the Bernstein operator are presented by the Ditzi an-Totik modulus of smoothness omega(phi)(2)(f, delta), whereby the st ep-weight phi is a function such that phi(2) is concave. The inverse d irection will be established for those step-weights phi for which phi( 2) and phi(2)/phi(2), phi(x) = root x(1-x), are concave functions. Thi s combines the classical estimate (phi = 1) and the estimate developed by Ditzian and Totik (phi = phi). In particular, the cases phi = phi( lambda),( )lambda is an element of [0, 1], are included.