Approximation on a disk. I
type: Item
instance of: scholarly article
title: Approximation on a disk. I
zbMATH Open document ID: 0789.30027
DOI: 10.1007/BF02571647
author: P. J. de Paepe
published in: Mathematische Zeitschrift
publication date: 1994-05-30T00:00:00Z
full work available at URL: https://eudml.org/doc/174476
review text: In the functions \(F\) and \(G\) are defined in a neighborhood of the origin in the complex plane and behave in a certain sense as \(z^ m\) and \(\overline z^ n\) with \(\text{gcd} (m,n)=1\), then on sufficiently small closed disks \(D\) around 0 every continuous function on \(D\) can be approximated uniformly on \(D\) by polynomials in \(F\) and \(G\). A new proof of this theorem is given. Our main result is an example showing that this approximation result is in general not valid for \(m=n=2\). In the process of constructing such an example one also comes across an example of two polynomially convex closed disks \(X_ 1\) and \(X_ 2\) in \(\mathbb{C}^ 2\) which meet at only one point, which are tangent at this point and are such that the union of the two disks is not polynomially convex. [For part II, see the following review].
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 32E20
zbMATH DE Number: 549722
MaRDI profile type: Publication
cites work: Analyticity on Rotation Invariant Families of Curves
cites work: Q5517147
cites work: Some Applications of the Stone-Weierstrass Theorem to Planar Rational Approximation
cites work: Approximation on Disks
cites work: On the polynomial convexity of the union of two maximal totally real subspaces of \(\mathbb C^n\)
cites work: Approximation on a disk
Wikidata QID: Q126270983
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description: scientific article; zbMATH DE number 549722