Whitney's theorem for local anisotropic polynomial L_p-approximation, 0<p<1

TL;DR

This paper extends Whitney's theorem for local anisotropic polynomial approximation in $L_p$ spaces to the case where 0 < p < 1, using a new proof method suitable for these quasi-normed spaces.

Contribution

It provides a new proof of Whitney's theorem for $L_p$ approximation with $0 < p < 1$, which was previously unproven with existing methods.

Findings

Established Whitney's theorem for $0 < p < 1$ in anisotropic polynomial approximation.

Characterized approximation error convergence rate via total mixed modulus of smoothness.

Extended the applicability of polynomial approximation theory to quasi-normed $L_p$ spaces.

Abstract

Dinh D\~ung and T. Ullrich have proven a multivariate Whitney's theorem for the local anisotropic polynomial approximation in $L_p(Q)$ for $1 \le p \le \infty$, where $Q$ is a $d$-parallelepiped in $\RR^d$ with sides parallel to the coordinate axes. They considered the error of best approximation of a function $f$ by algebraic polynomials of fixed degree at most $r_i - 1$ in variable $x_i,\ i=1,...,d$. The convergence rate of the approximation error when the size of $Q$ going to 0 is characterized by a so-called total mixed modulus of smoothness. The method of proof used by these authors is not suitable to the case $0 <p<1$. In the present paper, by a different method we proved this theorem for $0< p \le \infty$.