Real and complex analysis, 3rd ed.
Real and complex analysis, 3rd ed.
Full length article: Doubly universal Taylor series
Journal of Approximation Theory
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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.