Criterion of uniform approximability by harmonic functions on compact sets in $\mathbb R^3$

For a function continuous on a compact set $X\subset\mathbb R^3$ and harmonic inside $X$, we obtain a criterion of uniform approximability by functions harmonic in a neighborhood of $X$ in terms of the classical harmonic capacity. The proof is based on an improved localization scheme of A. G. Vitushkin, on a special geometric construction, and on the methods of the theory of singular integrals.