On uniform approximation by harmonic functions
type: Item
instance of: scholarly article
title: On uniform approximation by harmonic functions
zbMATH Open document ID: 0638.41032
published in: Michigan Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: Extending author's previous work on connections between approximation by analytic functions and the isoperimetric inequality [J. Funct. Anal. 58, 175-193 (1984; Zbl 0551.46032), J. Approximation Theory 50, 127-132 (1987; Zbl 0612.41024), Proc. Am. Math. Soc. 101, 475-483 (1987)] the concept of the harmonic content \(\Delta\) (\({\mathfrak X})=^{def}dist_{C({\mathfrak X})}(| X|\) 2,H(\({\mathfrak X}))\) is introduced in the paper. Here, \({\mathfrak X}\subset {\mathbb{R}}^ n \)is a compact set, \(| X|\) \(2=\sum^{n}_{1}X\) \(2_ i\), \(X=(X_ 1,...,X_ n)\in {\mathbb{R}}^ n \)and H(\({\mathfrak X})=^{def}\) closure of functions harmonic in a neighborhood of \({\mathfrak X}\) in the space C(X) of all continuous functions on \({\mathfrak X}\). It is shown that \(\Lambda (X)=0\Leftrightarrow\) H(\({\mathfrak X})=C({\mathfrak X})\) (the ``Stone- Weierstrass theorem for H(\({\mathfrak X})'')\). Also, \(\Lambda\) (\({\mathfrak X})\) is shown to enjoy a simple geometric estimate (\(\Lambda\) (\({\mathfrak X})\leq (1/2)R^ 2_{{\mathfrak X}}\), \(R_{{\mathfrak X}}\) is the ``volume radius'' of \({\mathfrak X}\) which is sharp (it becomes equality for balls). After this paper had been published, it was pointed out that the main theorem can also be derived from the results of \textit{W. Hansen} in fine potential theory [Ill. J. Math. 29, 103-107 (1985)] obtained, however, by completely different methods. Recently, the author generalized the above results to quite a general class of uniformly elliptic operators and also, obtained the sharp lower bounds for \(\Lambda\) (\({\mathfrak X}):\Lambda\) (\({\mathfrak X})\geq (1/2)\tilde R^ 2_ X\), where \(\tilde R_ X\) is the ``harmonic radius'' of \({\mathfrak X}\).
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 4040496
zbMATH Keywords: harmonic functions
zbMATH Keywords: Green's function
zbMATH Keywords: Hahn-Banach duality
zbMATH Keywords: analytic functions
zbMATH Keywords: isoperimetric inequality
zbMATH Keywords: harmonic content
zbMATH Keywords: fine potential theory
author: Dmitry Khavinson
reviewed by: Dmitry Khavinson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029003626
OpenAlex ID: W2015386933
DOI: 10.1307/MMJ/1029003626
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description: scientific article; zbMATH DE number 4040496