Universal Polynomial Expansions of Harmonic Functions |
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Authors: | Myrto Manolaki |
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Affiliation: | 1. School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
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Abstract: | Let Ω be a domain in ? N such that $left(mathbb{R}^{N}cuplbraceinftyrbraceright)setminusOmega$ is connected. It is known that, for each w?∈?Ω, there exist harmonic functions on Ω that are universal at w, in the sense that partial sums of their homogeneous polynomial expansion about w approximate all plausibly approximable functions in the complement of Ω. Under the assumption that Ω omits an infinite cone, it is shown that the connectedness hypothesis on $left(mathbb{R}^{N}cuplbraceinftyrbraceright)setminusOmega$ is essential, and that a harmonic function which is universal at one point is actually universal at all points of Ω. |
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