Universal Polynomial Expansions of Harmonic Functions

Abstract

Let Ω be a domain in ℝ^N such that $\left(\mathbb{R}^{N}\cup\lbrace\infty\rbrace\right)\setminus\Omega$ 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 $\left(\mathbb{R}^{N}\cup\lbrace\infty\rbrace\right)\setminus\Omega$ is essential, and that a harmonic function which is universal at one point is actually universal at all points of Ω.