Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update

TL;DR

This paper updates previous research on shape-preserving polynomial approximation, discussing recent uniform estimates, open problems, and differences in constrained approximation degrees, with new results on monotone functions.

Contribution

It provides recent uniform estimates in comonotone approximation, highlights differences in co-q-monotone cases, and establishes new limit relations for monotone function approximation.

Findings

Uniform estimates in comonotone approximation discussed.

Differences in co-q-monotone approximation cases highlighted.

Limit relations for monotone functions established.

Abstract

It is not surprising that one should expect that the degree of constrained (shape preserving) approximation be worse than the degree of unconstrained approximation. However, it turns out that, in certain cases, these degrees are the same. The main purpose of this paper is to provide an update to our 2011 survey paper. In particular, we discuss recent uniform estimates in comonotone approximation, mention recent developments and state several open problems in the (co)convex case, and reiterate that co-$q$-monotone approximation with $q\ge 3$ is completely different from comonotone and coconvex cases. Additionally, we show that, for each function $f$ from $\Delta^{(1)}$, the set of all monotone functions on $[-1,1]$, and every $\alpha>0$, we have \[ \limsup_{n\to\infty} \inf_{P_n\in\mathbb P_n\cap\Delta^{(1)}} \left\| \frac{n^\alpha(f-P_n)}{\varphi^\alpha} \right\| \le c(\alpha) \limsup_{n\to\infty} \inf_{P_n\in\mathbb P_n} \left\| \frac{n^\alpha(f-P_n)}{\varphi^\alpha} \right\| \] where $\mathbb P_n$ denotes the set of algebraic polynomials of degree $<n$, $\varphi(x):=\sqrt{1-x^2}$, and $c=c(\alpha)$ depends only on $\alpha$.