Universal Polynomial Expansions of Harmonic Functions

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.