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Universal polynomial expansions of harmonic functions. (English) Zbl 1271.31003
Let $$h$$ be a harmonic function on a domain $$\Omega \subset \mathbb{R}^{N}$$ and let $$w\in \Omega$$. We say that $$h\in U(\Omega ,w)$$ if, on each compact $$K\subset \mathbb{R}^{N}\backslash \Omega$$ with connected complement, arbitrary harmonic polynomials can be uniformly approximated by suitable partial sums of the homogeneous polynomial expansion of $$h$$ about $$w$$. It is known that $$U(\Omega ,w)\neq \emptyset$$ if $$(\mathbb{R}^{N}\cup \{\infty \})\backslash \Omega$$ is connected. The author shows, among other things, that, provided $$\Omega$$ omits an infinite cone, (1) $$U(\Omega ,w)\neq \emptyset$$ if and only if $$(\mathbb{R}^{N}\cup \{\infty \})\backslash \Omega$$ is connected; (2) the collection $$U(\Omega ,w)$$ is independent of the choice of $$w$$; (3) no function in $$U(\Omega ,w)$$ has a harmonic extension to a larger domain. The paper includes, and exploits, a careful analysis of the gap structure of the homogeneous polynomial expansion of functions belonging to $$U(\Omega ,w)$$ for domains $$\Omega$$ that omit a cone.

##### MSC:
 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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##### References:
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