Universal Overconvergence of Polynomial Expansions of Harmonic Functions

Abstract

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