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

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 ${\mathrm{L}\mathrm{i}\mathrm{p}}_{\omega }(K)$ 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 $\omega (\delta )\mathrm{/}\delta$ on the gradient holds.