Universal polynomial expansions of harmonic functions
type: Item
instance of: scholarly article
title: Universal polynomial expansions of harmonic functions
zbMATH Open document ID: 1271.31003
author: Myrto Manolaki
published in: Potential Analysis
publication date: 2013-05-06T00:00:00Z
review text: 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.
reviewed by: Stephen J. Gardiner
Mathematics Subject Classification ID: 31B05
zbMATH DE Number: 6160851
zbMATH Keywords: harmonic function
zbMATH Keywords: harmonic approximation
zbMATH Keywords: homogeneous harmonic polynomials
zbMATH Keywords: universal series
MaRDI profile type: Publication
full work available at URL: s11118-012-9303-z
OpenAlex ID: W1985537595
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DOI: 10.1007/S11118-012-9303-Z
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description: scientific article; zbMATH DE number 6160851