Harmonic approximation on compact sets
type: Item
instance of: scholarly article
title: Harmonic approximation on compact sets
zbMATH Open document ID: 1064.31006
DOI: 10.1007/BF03321042
published in: Computational Methods and Function Theory
publication date: 2004-11-01T00:00:00Z
review text: Let \(\Omega\) be a \(C^\infty\) oriented connected non-compact Riemannian manifold, and let \(X\), \(Y\) be compact subsets of \(\Omega\) such that \(X\subset Y\). This paper offers necessary and sufficient conditions on \(X\) and \(Y\), formulated in terms of thinness, so that every harmonic function on \(X\) (or, more generally, every continuous function on \(X\) that is harmonic on \(X^0\)) can be uniformly approximated on \(X\) by functions harmonic on \(Y\). These results generalise, using similar arguments, the known case where \(Y\) is the union of \(X\) and all the bounded components of \(\Omega\setminus X\). The results are then applied to the location of singularities in harmonic Runge-type approximation theorems.
reviewed by: Stephen J. Gardiner
Mathematics Subject Classification ID: 31C12
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 31A05
zbMATH DE Number: 2112943
zbMATH Keywords: harmonic function
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: uniform approximation
zbMATH Keywords: thin set
author: Thomas Bagby
author: P. M. Gauthier
MaRDI profile type: Publication
cites work: Uniform harmonic approximation on Riemannian manifolds
cites work: Q4309437
cites work: Simplicial cones in potential theory
cites work: Q5662251
cites work: Approximation et caractère de quasi-analyticité dans la théorie axiomatique des fonctions harmoniques
cites work: Potentiel fin et algèbres de fonctions analytiques. I
cites work: Finely harmonic functions
cites work: Q4836158
cites work: Global Approximation in Harmonic Spaces
cites work: Subharmonic Extensions and Approximations
cites work: Q5581926
cites work: A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations
cites work: Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution
cites work: Q3706198
description: scientific article; zbMATH DE number 2112943