A Runge type theorem for product of planar domains
type: Item
instance of: scholarly article
title: A Runge type theorem for product of planar domains
zbMATH Open document ID: 1380.32004
author: Javier Falcó
author: V. Nestoridis
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2017-11-01T00:00:00Z
review text: Let \(A(\Omega) \) be the Banach algebra of all functions \(f\) from \(\overline{\Omega}=\Pi_{i\in I}U_i\) to \(\mathbb C\) that are continuous on \(\overline{\Omega}\) with respect to the product topology and separately holomorphic in \(\Omega\), where \(I\) is an arbitrary set and \(U_i\) are planar domains of some type. The main result of this article: Theorem. Let \(U_i\), \(i\in I\), be an arbitrary family of bounded planar domains such that \(\mathrm{int}(\overline{U}_i)=U_i\) and \((\mathbb C\cup \{\infty\})\setminus \overline{U_i}\) has finitely many components \(V_j^i\), \(j=1, \ldots, n_i\), \(i\in I\). Let \(\alpha_i^j\in V^j_i \) be fixed. If \(\overline{\Omega}=\Pi_{i\in I}U_i\), then, for every function \(f\in A(\Omega)\) and every \(\epsilon>0\), there exists a function \(g\in A(\Omega)\) depending on a finite number of coordinates \(F\subset I\) such that \(||f-g||<\epsilon\) and the function \(g\) is a finite sum of finite products of rational functions of one variable \(z_i\), \(i\in F\), with poles in the set \(S_i=\{\alpha^j_i : j=1, \ldots, n_i,\}\), \(i\in F\).
Mathematics Subject Classification ID: 32A30
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 32A38
Mathematics Subject Classification ID: 46G20
zbMATH DE Number: 6802043
zbMATH Keywords: products of planar domains
zbMATH Keywords: rational approximation
zbMATH Keywords: infinite-dimensional holomorphy
reviewed by: Peter Dovbush
MaRDI profile type: Publication
full work available at URL: s13398-016-0353-8
OpenAlex ID: W2554489141
cites work: UNIVERSAL LAURENT SERIES
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cites work: Q3711155
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cites work: Uniform approximations by holomorphic functions
DOI: 10.1007/S13398-016-0353-8
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description: scientific article; zbMATH DE number 6802043