Universal Overconvergence of Polynomial Expansions of Harmonic Functions |
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Authors: | D H Armitage |
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Institution: | Department of Pure Mathematics, Queen's University Belfast, Belfast, BT7 1NN, Northern Irelandf1 |
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Abstract: | For each compact subset K of
N let
(K) denote the space of functions that are harmonic on some neighbourhood of K. The space
(K) is equipped with the topology of uniform convergence on K. Let Ω be an open subset of
N such that 0Ω and
N\Ω is connected. It is shown that there exists a series ∑Hn, where Hn is a homogeneous harmonic polynomial of degree n on
N, 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
(K) for every compact subset K of
N\Ω with connected complement. Some refinements are given and our results are compared with an analogous theorem concerning overconvergence of power series. |
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Keywords: | harmonic polynomial overconvergence series density universal |
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