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

For each q∈N0q∈N0, we construct positive linear polynomial approximation operators MnMn that simultaneously preserve kk-monotonicity for all 0≤k≤q0≤k≤q and yield the estimate|f(x)−Mn(f,x)|≤cω2φλ(f,n−1φ1−λ/2(x)(φ(x)+1/n)−λ/2), for x∈[0,1]x∈[0,1] and λ∈[0,2)λ∈[0,2), where φ(x):=x(1−x) and ω2ψ is the second Ditzian–Totik modulus of smoothness corresponding to the “step-weight function” ψψ. In particular, this implies that the rate of best uniform qq-monotone polynomial approximation can be estimated in terms of ω2φ(f,1/n).