Uniform approximation by entire functions that are all bounded on a given set
type: Item
instance of: scholarly article
title: Uniform approximation by entire functions that are all bounded on a given set
zbMATH Open document ID: 0920.30029
published in: Constructive Approximation
publication date: 1999-09-23T00:00:00Z
review text: The famous approximation theorem of Arakelyan (cf. \textit{D. Gaier}: Lectures on complex approximation [1980, Zbl 0442.30038)]) gives a complete (topological) description of those closed sets \(F\) in the complex plane which admit uniform approximation of functions \(f\in A(F)\) (i.e. continuous on \(F\) and holomorphic in the interior of \(F)\) by entire functions. The authors study how far this approximation can be done by entire functions which are bounded on another given closed set \(G\subset\mathbb{C}\). They prove that this can be done iff \(G\) is contained in some closed set \(G_1\neq\mathbb{C}\) with \(G\subset G_1\neq \mathbb{C}\) where \(F\) and \(G_1\) both fulfill the condition of Arakelyan's theorem and \(F\cap G_1\) is bounded.
reviewed by: Gerald Schmieder
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 1268074
Wikidata QID: Q56812626
author: A. A. Danielyan
author: Lee A. Rubel
MaRDI profile type: Publication
full work available at URL: s003659900085
OpenAlex ID: W2074117857
DOI: 10.1007/S003659900085
description: scientific article; zbMATH DE number 1268074