Pervasive algebras of analytic functions
type: Item
instance of: scholarly article
title: Pervasive algebras of analytic functions
zbMATH Open document ID: 0981.46044
author: M. Sanabria-García
author: Ivan Netuka
author: Anthony O'Farrell
published in: Journal of Approximation Theory
publication date: 2002-03-18T00:00:00Z
full work available at URL: OFarrellNo60.pdf
review text: Let \(X\) be a compact Hausdorff space and \(S\) a complex or real closed subspace of \(C(X,\mathbb{C})\) or \(C(X,\mathbb{R})\) respectively, and let \(Y\) be a closed subset of \(X\). \(S\) is said to be complex or real pervasive on \(Y\) if the functions of \(S\) restricted to \(E\) are dense in \(C(E,\mathbb{C})\) or \(C(E,\mathbb{R})\), respectively for each proper closed subset \(E\) of \(Y\). These properties are investigated for the case where \(X\) is an open proper subset \(U\) of the Riemann sphere \(\widehat{\mathbb{C}}\) and \(S\) is the algebra \(A(U)\) of all complex valued functions continuous on \(\widehat{\mathbb{C}}\) and analytic on \(U\), or \(S= \text{Re }A(U)\), respectively, and it is supposed that \(U\) has no inessential boundary points -- i.e. points where all functions of \(A(U)\) can be extended analytically. Then \(A(U)\) is complex pervasive on \(\partial U\) if and only if \(\partial U_i=\partial U\) for each component \(U_i\) of \(U\). If \(\text{Re }A(U)\) is real pervasive on \(\partial U\) then \(U\) has at most one component \(U_k\) that is not simply connected, and in this case \(\partial U_k=\partial U\). If on the other hand \(U\) has at least one component \(U_k\) such that \(\partial U_k=\partial U\) then \(\text{Re }A(U)\) is real pervasive on \(\partial U\). In case that all components \(U_i\) of \(U\) are simply connected and \(\partial U_i\neq\partial U\), then the real pervasiveness is characterized in terms of properties of the boundary points.
reviewed by: Gerhard Garske
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 30H05
zbMATH DE Number: 1569465
zbMATH Keywords: pervasive algebras of analytic functions
zbMATH Keywords: Riemann sphere
zbMATH Keywords: inessential boundary points
zbMATH Keywords: boundary points
MaRDI profile type: Publication
OpenAlex ID: W2050014449
Wikidata QID: Q116636823
cites work: Q5597848
cites work: Q5583306
cites work: Q3724984
cites work: Analytic Capacity and Approximation Problems
cites work: Q5615632
cites work: Pointwise bounded approximation and Dirichlet algebras
cites work: Maximal algebras of continuous functions
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DOI: 10.1006/JATH.2000.3496
description: scientific article; zbMATH DE number 1569465