On a problem of Dickmeis and Nessel concerning the approximation by Bernstein polynomials

For f is an element of C[0, 1] let B-n(f;x) denote the nth Bernstein polyno mial and omega*(f;t) the second modulus of smoothness. Continuing: the inve stigations by W. Dickmeis and R.J. Nessel it is shown that for each abstrac t modulus of continuity omega there exists a counterexample f(omega) is an element of C[0, 1] such that on the one hand omega*(f(omega); t) = O(omega( t)) and on the other hand lim sup(n-->infinity) \B-n(f(omega);x) - f(omega) (x)\/omega(x(1 - x)/n) greater than or equal to c > 0 simultaneously for al l x is an element of (0, 1). Furthermore, a pointwise lethargy assertion is established.