D. Leviatan

K. A. Kopotun, D. Leviatan, A. Prymak et al.

We survey developments, over the last thirty years, in the theory of Shape Preserving Approximation (SPA) by algebraic polynomials on a finite interval. In this article, "shape" refers to (finitely many changes of) monotonicity, convexity, or q-monotonicity of a function (for definition, see Section 4). It is rather well known that it is possible to approximate a function by algebraic polynomials that preserve its shape (i.e., the Weierstrass approximation theorem is valid for SPA). At the same time, the degree of SPA is much worse than the degree of best unconstrained approximation in some cases, and it is "about the same" in others. Numerous results quantifying this difference in degrees of SPA and unconstrained approximation have been obtained in recent years, and the main purpose of this article is to provide a "bird's-eye view" on this area, and discuss various approaches used. In particular, we present results on the validity and invalidity of uniform and pointwise estimates in terms of various moduli of smoothness. We compare various constrained and unconstrained approximation spaces as well as orders of unconstrained and shape preserving approximation of particular functions, etc. There are quite a few interesting phenomena and several open questions.

K. Kopotun, D. Leviatan, A. Prymak et al.

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 $ω_2^φ \left(f, 1/n \right)$.