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On directional Whitney inequality

Part of: General convexity Approximations and expansions

Published online by Cambridge University Press:  26 February 2021

Feng Dai  and
Andriy Prymak
Feng Dai*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, ABT6G 2G1, Canada
Andriy Prymak
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg, MBR3T 2N2, Canada e-mail: prymak@gmail.com
*
e-mail: fdai@ualberta.ca
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Abstract

This paper studies a new Whitney type inequality on a compact domain $\Omega \subset {\mathbb R}^d$ that takes the form

$$ \begin{align*} \inf_{Q\in \Pi_{r-1}^d(\mathcal{E})} \|f-Q\|_p \leq C(p,r,\Omega) \omega_{\mathcal{E}}^r(f,\mathrm{diam}(\Omega))_p,\ \ r\in {\mathbb N},\ \ 0<p\leq \infty, \end{align*} $$
where $\omega _{\mathcal {E}}^r(f, t)_p$ denotes the rth order directional modulus of smoothness of $f\in L^p(\Omega )$ along a finite set of directions $\mathcal {E}\subset \mathbb {S}^{d-1}$ such that $\mathrm {span}(\mathcal {E})={\mathbb R}^d$ , $\Pi _{r-1}^d(\mathcal {E}):=\{g\in C(\Omega ):\ \omega ^r_{\mathcal {E}} (g, \mathrm {diam} (\Omega ))_p=0\}$ . We prove that there does not exist a universal finite set of directions $\mathcal {E}$ for which this inequality holds on every convex body $\Omega \subset {\mathbb R}^d$ , but for every connected $C^2$ -domain $\Omega \subset {\mathbb R}^d$ , one can choose $\mathcal {E}$ to be an arbitrary set of d independent directions. We also study the smallest number $\mathcal {N}_d(\Omega )\in {\mathbb N}$ for which there exists a set of $\mathcal {N}_d(\Omega )$ directions $\mathcal {E}$ such that $\mathrm {span}(\mathcal {E})={\mathbb R}^d$ and the directional Whitney inequality holds on $\Omega $ for all $r\in {\mathbb N}$ and $p>0$ . It is proved that $\mathcal {N}_d(\Omega )=d$ for every connected $C^2$ -domain $\Omega \subset {\mathbb R}^d$ , for $d=2$ and every planar convex body $\Omega \subset {\mathbb R}^2$ , and for $d\ge 3$ and every almost smooth convex body $\Omega \subset {\mathbb R}^d$ . For $d\ge 3$ and a more general convex body $\Omega \subset {\mathbb R}^d$ , we connect $\mathcal {N}_d(\Omega )$ with a problem in convex geometry on the X-ray number of $\Omega $ , proving that if $\Omega $ is X-rayed by a finite set of directions $\mathcal {E}\subset \mathbb {S}^{d-1}$ , then $\mathcal {E}$ admits the directional Whitney inequality on $\Omega $ for all $r\in {\mathbb N}$ and $0<p\leq \infty $ . Such a connection allows us to deduce certain quantitative estimate of $\mathcal {N}_d(\Omega )$ for $d\ge 3$ .

A slight modification of the proof of the usual Whitney inequality in literature also yields a directional Whitney inequality on each convex body $\Omega \subset {\mathbb R}^d$ , but with the set $\mathcal {E}$ containing more than $(c d)^{d-1}$ directions. In this paper, we develop a new and simpler method to prove the directional Whitney inequality on more general, possibly nonconvex domains requiring significantly fewer directions in the directional moduli.

Keywords

Whitney-type inequalitydirectional modulus of smoothnessconvex domainsC2-domainsmultivariate polynomialsillumination of convex bodiesX-ray numberillumination number

MSC classification

Primary: 41A10: Approximation by polynomials
Secondary: 41A25: Rate of convergence, degree of approximation 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section) 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 52A40: Inequalities and extremum problems
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 3 , June 2022 , pp. 833 - 857
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

The first author was supported by NSERC of Canada Discovery grant RGPIN-2020-03909, and the second author was supported by NSERC of Canada Discovery grant RGPIN-2020-05357.

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