Yet another look at positive linear operators, [formula omitted]-monotonicity and applications

Abstract

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)|≤cω2φλ(f,n−1φ1−λ/2(x)(φ(x)+1/n)−λ/2), for x∈[0,1] and 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” ψ. In particular, this implies that the rate of best uniform q-monotone polynomial approximation can be estimated in terms of ω2φ(f,1/n).