Universal overconvergence of polynomial expansions of harmonic functions

For each compact subset K of R-N let H(K) denote the space of functions that are harmonic on some neighbourhood of K. The space H(K) is equipped with the topology of uniform convergence on K. Let Omega be an open subset of R-N such that 0 epsilon Omega and R-N\Omega is connected. It is shown that there exists a series E H-n where H-n is a homogeneous harmonic polynomial of degree n on R-N such that (i) Sigma H-n converges on some ball of centre 0 to a function that is continuous on Omega and harmonic on Omega, (ii) the partial sums of Sigma H-n are dense in H(K) for every compact subset K of R-N\Omega with connected complement. Some refinements are given and our results are compared with an analogous theorem concerning overconvergence of power series. (C) 2002 Elsevier Science (USA).