\(\text{Lip }\alpha\) harmonic approximation on closed sets (Q2716118)

Let \(G\) be a domain in \(\mathbb{C}\) and \(F\) be a relatively closed subset of \(G\). The reviewer [Ann. Inst. Fourier 44, No. 1, 65-91 (1994; Zbl 0795.31004)] has (in particular) characterized those pairs \((G,F)\) such that any function \(u\) which is continuous on \(F\) and harmonic on \(F^0\) can be uniformly approximated on \(F\) by harmonic functions \(v\) on \(G\). The present paper establishes an analogue of this result for \(\text{Lip }\alpha\) approximation \((0<\alpha< 1/2)\) with the same topological characterization as in the uniform case. This extends to closed sets a theorem of \textit{P. V. Paramonov} [Mat. Sb. 184, No. 2, 105-128 (1993; Zbl 0851.41029)] concerning \(\text{Lip }\alpha\) approximation on compact sets by harmonic polynomials. The authors also consider the behaviour of the error of approximation \((v-u)\) near the boundary of \(G\).