A note on Mergelyan's theorem
type: Item
instance of: scholarly article
title: A note on Mergelyan's theorem
zbMATH Open document ID: 0537.30039
author: Cho-ichiro Matsuoka
published in: Journal of Mathematics of Kyoto University
publication date: 1983-01-01T00:00:00Z
review text: Suppose that X is a compact subset of the complex plane and that the complement of X is connected. Let A(X) denote the set of functions which are continuous on X and analytic on the interior of X. Let R(X) be the subset of A(X) consisting of uniform limits of polynomials. Mergelyan's theorem states that \(R(X)=A(X)\) [\textit{S. N. Mergelyan}, Dokl. Akad. Nauk SSSR, N. Ser. 78, 405-408 (1951; Zbl 0042.082)]. The author gives a direct proof of this theorem based on analysis of the associated spaces of annihilating measures. His proof uses, among other things, ideas of \textit{E. Bishop} [Duke Math. J. 27, 331-340 (1960; Zbl 0094.272)].
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 3853457
zbMATH Keywords: Mergelyan's theorem
reviewed by: Richard Rochberg
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250521556
OpenAlex ID: W1522563226
DOI: 10.1215/KJM/1250521556
description: scientific article; zbMATH DE number 3853457