Difference spectrum and spectral synthesis
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.
scientific article; zbMATH DE number 1321874
scientific article