Universal polynomial expansions of harmonic functions
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.
scientific article; zbMATH DE number 6160851
scientific article