Universal Overconvergence of Polynomial Expansions of Harmonic Functions Original Research Article

For each compact subset K of RN 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 Ω be an open subset of RN such that 0∈Ω and RN\Ω is connected. It is shown that there exists a series ∑Hn, where Hn is a homogeneous harmonic polynomial of degree n on RN, such that (i) ∑Hn converges on some ball of centre 0 to a function that is continuous on Ω and harmonic on Ω, (ii) the partial sums of ∑Hn are dense in H(K) for every compact subset K of RN\Ω with connected complement. Some refinements are given and our results are compared with an analogous theorem concerning overconvergence of power series.