Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update
The SMAI Journal of computational mathematics, Volume S5 (2019), pp. 99-108.

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 ${\Delta }^{\left(1\right)}$, the set of all monotone functions on $\left[-1,1\right]$, and every $\alpha >0$, we have

 $\underset{n\to \infty }{lim sup}\underset{{P}_{n}\in {ℙ}_{n}\cap {\Delta }^{\left(1\right)}}{inf}∥\frac{{n}^{\alpha }\left(f-{P}_{n}\right)}{{\varphi }^{\alpha }}∥\le c\left(\alpha \right)\underset{n\to \infty }{lim sup}\underset{{P}_{n}\in {ℙ}_{n}}{inf}∥\frac{{n}^{\alpha }\left(f-{P}_{n}\right)}{{\varphi }^{\alpha }}∥$

where ${ℙ}_{n}$ denotes the set of algebraic polynomials of degree $, $\varphi \left(x\right):=\sqrt{1-{x}^{2}}$, and $c=c\left(\alpha \right)$ depends only on $\alpha$.

Published online:
DOI: 10.5802/smai-jcm.54
Classification: 41A10, 41A17, 41A25
Keywords: Approximation by algebraic polynomials, shape preserving approximation, constrained approximation

K. A. Kopotun 1; D. Leviatan 2; I. A. Shevchuk 3

1 Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada
2 Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6139001, Israel
3 Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, 01033 Kyiv, Ukraine
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K. A. Kopotun; D. Leviatan; I. A. Shevchuk. Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update. The SMAI Journal of computational mathematics, Volume S5 (2019), pp. 99-108. doi : 10.5802/smai-jcm.54. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.54/

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