On harmonic approximation of Lipschitz functions on compacts in $\mathbb{R}^d$

TL;DR

This paper establishes a quantitative relationship between Lipschitz functions on porous compact sets in Euclidean space and their harmonic approximations, extending classical approximation theorems with explicit rates.

Contribution

It proves a Jackson-Bernstein type theorem linking Lipschitz classes and harmonic approximation rates on porous sets in $\,\mathbb{R}^d$.

Findings

Characterization of Lipschitz functions via harmonic approximation rates

Explicit approximation rates depending on the continuity modulus

Extension of classical approximation theorems to porous sets

Abstract

Given a porous compact $K \subset \mathbb{R}^d$ and a continuity modulus $\omega$, we prove a quantitative Jackson-Bernstein type theorem on harmonic approximation. That is, a function $f$ belongs to the class $\mathrm{Lip}_{\omega}(K)$ if and only if $f$ can be approximated uniformly on $K$ with a rate of $\omega(\delta)$ by a function that is harmonic in the $\delta$-neighborhood of $K$, provided the uniform estimate $\omega(\delta)/\delta$ on the gradient holds.