On harmonic approximation in the $C^1$-norm

A criterion is established for the possibility of approximation by harmonic functions and, in particular, by harmonic polynomials in the $C^1$-norm on compact subsets of $\mathbf R^n$. This criterion, which is in terms of harmonic $C^1$-capacity in $\mathbf R^n$, yields a natural analog to the theorem of Vitushkin on rational approximation in terms of analytic capacity.