In this paper we g i v e some necessary and su cient conditions for uniform approximability of functions by polyanalytic polynomials on plane compact sets of special form. Also connections with the corresponding Dirichlet problem are considered.

We shall study geometric properties of the harmonic Lipjcapacity κ! n (E), E C R n . It is related to functions which are harmonic outside E and locally Lipschitzian everywhere. We shall show that κ! n+1 {E x /) is comparable to κ' n {E) for EcR n and for intervals / C R. We shall also show that if E lies on a Lipschitz graph, then κ' n (E) is comparable to the (nl)-dimensional Hausdorff measure Ή n~1 (E). Finally we give some general criteria to guarantee that κ! n {E) -0 although n n~ι {E) >o.

A criterion is established for the possibility of approximation by harmonic functions and, in particular, by harmonic polynomials in the С'-norm on compact subsets of R" . This criterion, which is in terms of harmonic С'-capacity in R" , yields a natural analog to the theorem of Vitushkin on rational approximation in terms of analytic capacity.

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