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

Abstract

For each $q\in{\mathbb{N}}_0$, we construct positive linear polynomial approximation operators $M_n$ that simultaneously preserve $k$-monotonicity for all $0\leq k\leq q$ and yield the estimate \[ |f(x)-M_n(f, x)| \leq c ω_2^{φ^λ} \left(f, n^{-1} φ^{1-λ/2}(x) \left(φ(x) + 1/n \right)^{-λ/2} \right) , \] for $x\in [0,1]$ and $λ\in [0, 2)$, where $φ(x) := \sqrt{x(1-x)}$ and $ω_2^ψ$ 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 $ω_2^φ \left(f, 1/n \right)$.