Uniform and Pointwise Shape Preserving Approximation by Algebraic Polynomials

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.Comment: 51 pages, 49 tables, survey, published in Surveys in Approximation Theory, 6 (2011), 24-7

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

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 $\alpha$

Degree of Simultaneous Coconvex Polynomial Approximation

: Let f 2 C 1 [\Gamma1; 1] change its convexity finitely many times in the interval, say s times, at Y s : \Gamma1 ! y s ! \Delta \Delta \Delta ! y 1 ! 1. We estimate the degree of simultaneous approximation of f and its derivative by polynomials of degree n, which change convexity exactly at the points Y s , and their derivatives. We show that provided n is sufficiently large, depending on the location of the points Y s , the rate of approximation can be estimated by C(s)=n times the second Ditzian--Totik modulus of smoothness of f 0 . This should be compared to a recent paper by the authors together with I. A. Shevchuk where f is merely assumed to be continuous and estimates of coconvex approximation are given by means of the third Ditzian--Totik modulus of smoothness. However, no simultaneous approximation is given there. 1991 Mathematics Subject Classification: 41A10, 41A17, 41A25, 41A29 Keywords: Coconvex polynomial approximation, Jackson estimates 1 Introduction and main re..