Difference spectrum and spectral synthesis
type: Item
instance of: scholarly article
title: Difference spectrum and spectral synthesis
zbMATH Open document ID: 0930.43004
author: Thettath K. Muraleedharan
author: K. Parthasarathy
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2000-02-03T00:00:00Z
review text: Consider the Fourier algebra \(A(G) \cong {\mathcal F}L^1(\widehat{G})\), \(G\) a locally compact abelian group. The \text {difference spectrum} of two ideals in \(A(G)\) is the set of all points of \(G\) where the ideals are locally different. This notion was introduced in papers by the reviewer [Harmonic analysis. Proc. Conf., Iraklion 1978, Lect. Notes Math. 781, 194-203 (1980; Zbl 0429.43008)] and by \textit{D. L. Salinger} and the reviewer [J. Lond. Math. Soc. (3) 26, 531-540 (1982; Zbl 0496.43004)]. It can be used as a means for describing the family of all closed ideals with cospectrum (`hull') \(E\), where \(E\) is a set of non--synthesis. In the present paper some results on unions, intersections, and Cartesian products of sets of synthesis are discussed, using the difference spectrum. A generalization is the \(n\)-difference spectrum (where \(n\) is a natural number) and the corresponding notion of \(n\)-weak synthesis (for convenience I choose a slightly different terminology). Here various results -- new ones, or improvements of existing ones -- are obtained. A sample result is the following: if the \(m\)-difference spectrum of a closed set \(E\) is contained in a closed subset of \(E\) of \(n\)--weak synthesis, then \(E\) is of \((n+m)\)-weak synthesis.
reviewed by: Jan Stegeman
Mathematics Subject Classification ID: 43A45
Mathematics Subject Classification ID: 43A46
zbMATH DE Number: 1321874
zbMATH Keywords: difference spectrum
zbMATH Keywords: spectral synthesis
zbMATH Keywords: weak spectral synthesis
zbMATH Keywords: Fourier algebra
zbMATH Keywords: locally compact abelian group
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178224853
OpenAlex ID: W2081285677
cites work: Q5615626
cites work: On Unions and Intersections of Sets of Synthesis
cites work: Nonsynthesizable Intersections of Sets of Synthesis
cites work: On weak spectral synthesis
cites work: Q5550080
cites work: Spectral synthesis for the Kronecker sets
cites work: Difference Spectra of Ideals for Non-Metrizable Groups
cites work: Q3866604
cites work: Weak Spectral Synthesis
cites work: Q3829997
DOI: 10.2748/TMJ/1178224853
description: scientific article; zbMATH DE number 1321874