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

For each q ∈ N0, we construct positive linear polynomial approximation operators Mn that simultaneously preserve k-monotonicity for all 0 ≤ k ≤ q and yield the estimate |f(x)−Mn(f, x)| ≤ cω λ 2 ( f, nφ(x) (φ(x) + 1/n) ) , for x ∈ [0, 1] and λ ∈ [0, 2), where φ(x) := √ x(1− x) and ω 2 is the second Ditzian-Totik modulus of smoothness corresponding to the “step-weight function” ψ. In particular, this implies that the rate of best uniform q-monotone polynomial approximation can be estimated in terms of ω 2 (f, 1/n).