TANGENTIAL APPROXIMATION IN HARMONIC SPACES

Let Omega be a strong Brelot harmonic space possessing a positive potential. If A subset of or equal to Omega, let H(A) (resp. I-C(A)) be the collection of all functions which are harmonic (resp. continuous and superharmonic) on an open set containing A. The main result asserts that the following three conditions on a closed subset E of Omega are equivalent: (a) (resp. (b)) for each u in C(E) boolean AND H(($) over circle E) (resp. I-C(E) boolean AND (c)(($) over circle E)) and each continuous function epsilon : E --> (0,1], there exists v in H(E) (resp. I-C(E)) such that 0 < v - u < epsilon on E; (c) (i) Omega\E and Omega\($) over circle E are thin at the same points, and (ii) for each compact set K there is a compact set L which contains all the connected components of ($) over circle E which intersect K.