Approximation of analytic functions by rational functions in Banach spaces
type: Item
instance of: scholarly article
title: Approximation of analytic functions by rational functions in Banach spaces
zbMATH Open document ID: 0564.46039
DOI: 10.1016/0022-1236(84)90090-9
published in: Journal of Functional Analysis
publication date: 1984-01-01T00:00:00Z
review text: The author proves several resuls on the rational approximation of analytic functions over certain compact subsets of a Banach space E with the approximation property. An important tool in the proof of these results is a generalized version of the Shilov-Arens-Calderón- Waelbroeck theorem. As an application of this result a simple proof of the Oka-Weil theorem in E is given. This proof allows the athor to state a result on approximation of continuous functions by polynomials over totally disconnected polynomially convex compact subsets of E. A vector- valued version of the Oka-Weil theorem is proved, and the author shows that the vector Oka-Weil property for a Banach space implies the approximation property. The rational functions on an open subset U of E are introduced, and the normal envelope of rationality \(U_{{\mathcal R}}\) is constructed as a subset of E. These results are fundamental for the part where the author proves Oka-Weil type theorems concerning the approximation of continuous functions by rational functions. Some results relating rational, polynomial and holomorphic convexity are proved. When E has the bounded approximation property, the author shows that \(U_{{\mathcal R}}\) is a domain of holomorphy.
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 32E30
Mathematics Subject Classification ID: 32D05
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 3899573
zbMATH Keywords: rational approximation of analytic functions over certain compact subsets of a Banach space
zbMATH Keywords: approximation property
zbMATH Keywords: Shilov-Arens-Calderón- Waelbroeck theorem
zbMATH Keywords: polynomials over totally disconnected polynomially convex compact subsets
zbMATH Keywords: vector-valued version of the Oka-Weil theorem
zbMATH Keywords: vector Oka-Weil property
zbMATH Keywords: envelope of rationality
zbMATH Keywords: rational, polynomial and holomorphic convexity
zbMATH Keywords: domain of holomorphy
author: Mário Carvalho Matos
MaRDI profile type: Publication
full work available at URL: 0022-1236(84)90090-9
OpenAlex ID: W2074882777
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description: scientific article; zbMATH DE number 3899573