Yet another look at positive linear operators, q-monotonicity and applications

For each q is an element of N-0, we construct positive linear polynomial approximation operators M-n that simultaneously preserve k-monotonicity for all 0 <= k <= q and yield the estimate vertical bar f(x) - M-n(f, x)vertical bar <= c omega(phi lambda)(2) (f,n(-1)phi(1-lambda/2)(x) (phi(x) + 1/n)(-lambda/2)), for x is an element of [0, 1] and lambda is an element of [0, 2), where phi(x) := root x(1-x) and omega(psi)(2) is the second Ditzian-Totik modulus of smoothness corresponding to the "step-weight function" psi. In particular, this implies that the rate of best uniform q-monotone polynomial approximation can be estimated in terms of omega(psi)(2) (f, 1/n). (C) 2016 Elsevier Inc. All rights reserved.