The failure of spectral synthesis on some types of discrete abelian groups
type: Item
instance of: scholarly article
title: The failure of spectral synthesis on some types of discrete abelian groups
zbMATH Open document ID: 1051.43002
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-06-15T00:00:00Z
review text: The setting is the space \({\mathcal C}(G)\) of complex-valued functions on a discrete Abelian group \(G\). The topology is that of pointwise convergence. The question of synthesis is ``Does every non-zero closed translation invariant subspace contain sufficient exponential monomials for their linear hull to be dense?'' (An exponential monomial is the product of a homomorphism into the multiplicative group of non-zero complex numbers by a homomorphism into the additive group of complex numbers.) The question was thought to have been settled in the affirmative for arbitrary discrete abelian groups by \textit{R. J. Elliot} [Proc. Camb. Philos. Soc. 61, 617--620 (1965; Zbl 0138.07402)], but the author of the present paper shows that synthesis fails on any Abelian group with torsion-free rank at least \(\omega\).
reviewed by: D. L. Salinger
Mathematics Subject Classification ID: 43A45
zbMATH DE Number: 2075657
zbMATH Keywords: spectral synthesis
zbMATH Keywords: discrete Abelian group
author: László Székelyhidi
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.11.041
OpenAlex ID: W2005068001
cites work: A representation theorem for $(X_1-1)(X_2-1)...(X_n-1)$ and its applications
cites work: Q5512375
cites work: Q3254054
cites work: Théorie générale des fonctions moyenne-périodiques
cites work: Q4002300
cites work: On the extension of exponential polynomials
cites work: Q5459512
DOI: 10.1016/J.JMAA.2003.11.041
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description: scientific article; zbMATH DE number 2075657