Stray, Arne Decomposition of approximable functions. (English) Zbl 0568.30035 Ann. Math. (2) 120, 225-235 (1984). The purpose of this paper is to give a description of the space \(A_ D(F)\) of functions on a relatively closed subset F of an open plane set D which can be approximated uniformly on F by functions in H(D), i.e. functions analytic on D. The main result is the decomposition \[ A_ D(F)=C_{ua}(F\cup \Omega (F))+H(D)). \] Here \(\Omega (F)=D\setminus (F\cup M_ F)\), where \(M_ F\) is the set of all \(z\in D\setminus F\) which can be joined to \(D^*\setminus D\) by an arc in \(D^*\setminus F\) \((D^*\) is the one point compactification of D) and \(C_{ua}(F\cup \Omega (F))\) denotes the uniformly continuous functions on \(F\cup \Omega (F)\) which are analytic in the interior of \(F\cup \Omega (F).\) Basic ingredients in the proof are Vitushkin’s scheme for rational approximation [see e.g. T. W. Gamelin, Uniform Algebras (1969; Zbl 0213.40401)], together with Arakelyan’s noncompact versions of Mergelyan’s classical polynomial approximation theorem [see e.g. D. Gaier: Vorlesungen über Approximationen im Komplexen (1980; Zbl 0442.30038)] and earlier related results by the author. Reviewer: B.Øksendal Cited in 2 ReviewsCited in 3 Documents MSC: 30E10 Approximation in the complex plane Keywords:uniform approximation on a relatively closed subset; Vitushkin’s scheme for rational approximation Citations:Zbl 0213.40401; Zbl 0442.30038 PDF BibTeX XML Cite \textit{A. Stray}, Ann. Math. (2) 120, 225--235 (1984; Zbl 0568.30035) Full Text: DOI OpenURL