Rational approximation on the unit sphere in \(\mathbb{C}^2\)
type: Item
instance of: scholarly article
title: Rational approximation on the unit sphere in \(\mathbb{C}^2\)
zbMATH Open document ID: 1067.32011
author: John T. Anderson
author: Alexander J. Izzo
author: John Wermer
published in: Michigan Mathematical Journal
publication date: 2004-11-11T00:00:00Z
review text: Let \(X\) be a compact subset of \(\mathbb C^n\) and let \(R(X)\) be the closure in \(C(X)\) of the set of rational functions holomorphic in a neighborhood of \(X\). The authors are interested in finding conditions on \(X\) which imply \(R(X)=C(X)\). Let \(\hat{X_r}=\{z\in\mathbb C^n:\;\text{every polynomial}\;Q\;\text{with}\;Q(z)=0\;\text{vanishes at some point of}\;X\}\). The condition \(X=\hat{X_r}\) (i.e. that \(X\) is rationally convex) is necessary for \(R(X)=C(X)\). If \(n=1\), every compact set \(X\) is rationally convex. However, there are known examples of sets without interior for which \(R(X)\neq C(X)\) (e.g. the Alice Roth's Swiss cheese). On the other hand, in virtue of the Hartogs-Rosenthal theorem, if the two-dimensional Lebesgue measure of \(X\) is zero, then \(R(X)=C(X)\). \textit{R. F. Basener} [Trans. Am. Math. Soc. 182, 353--381 (1973; Zbl 0239.46051)] has given examples of rationally convex sets \(X\subset\mathbb C^2\) of the form \(\{(z,w)\in\partial B:\;z\in E\}\), where \(E\subset\mathbb C\) is a suitable Swiss cheese, such that \(R(X)\neq C(X)\). These sets have the property that \(\sigma(X)>0\), where \(\sigma\) is three-dimensional Hausdorff measure on \(\partial B\). In view of the one-dimensional Hartogs-Rosenthal theorem, it is reasonable to conjecture that if \(X=\hat{X_r}\) and \(\sigma(X)=0\), then \(R(X)=C(X)\). In the paper, the authors give several contributions to the study of this question.
reviewed by: Wiesław Pleśniak
Mathematics Subject Classification ID: 32E30
Mathematics Subject Classification ID: 41A20
zbMATH DE Number: 2114458
zbMATH Keywords: rational approximation
zbMATH Keywords: Hartogs-Rosenthal theorem
zbMATH Keywords: rational convexity
zbMATH Keywords: Henkin transform
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1080837737
OpenAlex ID: W2004023502
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DOI: 10.1307/MMJ/1080837737
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description: scientific article; zbMATH DE number 2114458