Yet another look at positive linear operators, $q$-monotonicity and applications
This provides a unified theoretical framework for monotone polynomial approximation, but the result is incremental as it extends known constructions for specific q to arbitrary q.
The authors construct positive linear polynomial operators that preserve k-monotonicity for all k up to q and achieve a pointwise approximation rate expressed via the Ditzian-Totik modulus of smoothness, leading to an estimate for the best uniform q-monotone polynomial approximation rate.
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 $ω_2^φ \left(f, 1/n \right)$.