On the closure of the sum of two uniform algebras on compact sets in \(\mathbb C\)
type: Item
instance of: scholarly article
title: On the closure of the sum of two uniform algebras on compact sets in \(\mathbb C\)
zbMATH Open document ID: 1237.46040
DOI: 10.1134/S0001434611010056
author: A. B. Zaitsev
published in: Mathematical Notes
publication date: 2011-11-25T00:00:00Z
review text: Let \(X\subseteq \mathbb C\) be a compact set without interior points and whose complement is disconnected. The polynomial convex hull of \(X\) is denoted by \(\widehat X\). For a homeomorphism \(\psi\) of \(\mathbb C\) let \(Z(X,\psi)\) be the closure in \(C(X)\) of the set of functions of the form \(p+(h\circ \psi)\) where \(p\) is a polynomial and \(h\) a rational function whose poles belong to the bounded components of \(\mathbb C\setminus\psi(X)\). The author shows the following result. Suppose that \(\psi\) is an orientation-preserving diffeomorphism of \(\mathbb C\) and that each point of the set \[ X\cap \overline{\left(\widehat X\right)^\circ} \] is a boundary point for some bounded component of \(\mathbb C\setminus X\). Then \(Z(X,\psi)=C(X)\). Several nice examples round up the paper.
reviewed by: Raymond Mortini
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 5980744
zbMATH Keywords: uniform approximation
zbMATH Keywords: polynomials
zbMATH Keywords: rational functions
zbMATH Keywords: homeomorphisms
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5980744