Polynomial approximation on $C^2$-domains

TL;DR

This paper develops new tools to measure smoothness of functions on $C^2$-domains and establishes direct and inverse polynomial approximation inequalities in various $L_p$ spaces, advancing approximation theory on complex domains.

Contribution

Introduces a computable modulus of smoothness for multivariate polynomial approximation on $C^2$-domains, proving Jackson and inverse inequalities with new localized polynomial partitions.

Findings

Established Jackson inequality for all $0<p extless=ty$

Proved inverse inequality for $1 extless=p extless=ty$

Developed localized polynomial partitions of unity on $C^2$-domains

Abstract

We introduce appropriate computable moduli of smoothness to characterize the rate of best approximation by multivariate polynomials on a connected and compact $C^2$-domain $\Omega\subset \mathbb{R}^d$. This new modulus of smoothness is defined via finite differences along the directions of coordinate axes, and along a number of tangential directions from the boundary. With this modulus, we prove both the direct Jackson inequality and the corresponding inverse for the best polynomial approximation in $L_p(\Omega)$. The Jackson inequality is established for the full range of $0<p\leq \infty$, while its proof relies on a recently established Whitney type estimates with constants depending only on certain parameters; and on a highly localized polynomial partitions of unity on a $C^2$-domain which is of independent interest. The inverse inequality is established for $1\leq p\leq \infty$, and its proof relies on a recently proved Bernstein type inequality associated with the tangential derivatives on the boundary of $\Omega$. Such an inequality also allows us to establish the inverse theorem for Ivanov's average moduli of smoothness on general compact $C^2$-domains.