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

Kopotun, K. A.; Leviatan, D.; Shevchuk, I. A.

The SMAI Journal of computational mathematics, Tome 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 $[-1,1]$, and every $\alpha >0$, we have

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

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

Classification : 41A10, 41A17, 41A25
Mots clés : Approximation by algebraic polynomials, shape preserving approximation, constrained approximation

[1] Bezkryla, S. I. 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] DeVore, R. A.; Lorentz, G. G. Constructive approximation, Grundlehren der Mathematischen Wissenschaften, 303, Springer, 1993, x+449 pages | MR | Zbl

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

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

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

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

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

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

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

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

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

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

[13] Leviatan, D.; Shevchuk, I. A. 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

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

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