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: https://doi.org/10.5802/smai-jcm.54
Classification: 41A10,  41A17,  41A25
Keywords: Approximation by algebraic polynomials, shape preserving approximation, constrained approximation
@article{SMAI-JCM_2019__S5__99_0,
author = {K. A. Kopotun and D. Leviatan and I. A. Shevchuk},
title = {Uniform and pointwise shape preserving approximation {(SPA)} by algebraic polynomials: an update},
journal = {The SMAI journal of computational mathematics},
pages = {99--108},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {S5},
year = {2019},
doi = {10.5802/smai-jcm.54},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.54/}
}
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/

[1] S. I. Bezkryla On Jackson–Stechkin type estimates for piecewise q-convex approximation of functions, Visnyk. Math. Mech., Kyiv. Univ. Im. Tarasa Shevchenka, Volume 36 (2016), pp. 6-10

[2] R. A. DeVore; G. G. Lorentz Constructive approximation, Grundlehren der Mathematischen Wissenschaften, Volume 303, Springer, 1993, x+449 pages | MR 1261635 | Zbl 0797.41016

[3] R. A. DeVore; X. M. Yu Pointwise estimates for monotone polynomial approximation, Constr. Approx., Volume 1 (1985) no. 4, pp. 323-331 | Article | MR 891762 | Zbl 0583.41006

[4] V. K. Dzyadyk; I. A. Shevchuk Theory of uniform approximation of functions by polynomials, Walter de Gruyter, 2008, xvi+480 pages

[5] H. H. Gonska; D. Leviatan; I. A. Shevchuk; H.-J. Wenz Interpolatory pointwise estimates for polynomial approximation, Constr. Approx., Volume 16 (2000) no. 4, pp. 603-629 | Article | MR 1771698 | Zbl 0976.41018

[6] K. A. Kopotun; D. Leviatan; A. Prymak; I. A. Shevchuk Uniform and pointwise shape preserving approximation by algebraic polynomials, Surv. Approx. Theory, Volume 6 (2011), pp. 24-74 | MR 2832606 | Zbl 1296.41001

[7] K. A. Kopotun; D. Leviatan; I. A. Shevchuk Interpolatory pointwise estimates for monotone polynomial approximation, J. Math. Anal. Appl., Volume 459 (2018) no. 2, pp. 1260-1295 | Article | MR 3732586 | Zbl 1377.41001

[8] K. A. Kopotun; D. Leviatan; I. A. Shevchuk Interpolatory estimates for convex piecewise polynomial approximation, J. Math. Anal. Appl., Volume 474 (2019) no. 1, pp. 467-479 | Article | MR 3912911 | Zbl 1414.41004

[9] D. Leviatan Pointwise estimates for convex polynomial approximation, Proc. Am. Math. Soc., Volume 98 (1986) no. 3, pp. 471-474 | Article | MR 857944 | Zbl 0617.41009

[10] D. Leviatan; D. V. Radchenko; I. A. Shevchuk Positive results and counterexamples in comonotone approximation, Constr. Approx., Volume 36 (2012) no. 2, pp. 243-266 | Article | MR 2957310 | Zbl 1254.41006

[11] D. Leviatan; I. A. Shevchuk Counterexamples in convex and higher order constrained approximation, East J. Approx., Volume 1 (1995) no. 3, pp. 391-398 | MR 1404356 | Zbl 0846.41006

[12] D. Leviatan; I. A. Shevchuk Comparing the degrees of unconstrained and shape preserving approximation by polynomials, J. Approximation Theory, Volume 211 (2016), pp. 16-28 | Article | MR 3547629 | Zbl 1353.41003

[13] D. Leviatan; I. A. Shevchuk Jackson type estimates for piecewise $q$-monotone approximation, $q\ge 3$, are not valid, Pure Appl. Funct. Anal., Volume 1 (2016) no. 1, pp. 85-96 | Zbl 1347.41007

[14] D. Leviatan; I. A. Shevchuk; O. V. Vlasiuk Positive results and counterexamples in comonotone approximation II, J. Approximation Theory, Volume 179 (2014), pp. 1-23 | Article | MR 3148883 | Zbl 1292.41003

[15] G. G. Lorentz; K. L. Zeller Degree of approximation by monotone polynomials. II, J. Approximation Theory, Volume 2 (1969), pp. 265-269 | Article | MR 244677 | Zbl 0175.06102