Geodesic
Characterisation of Locally Compact Abelian Groups Having Spectral Synthesis
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e145

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper we solve a long-standing problem which goes back to Laurent Schwartz’s work on mean periodic functions. Namely, we completely characterize those locally compact Abelian groups having spectral synthesis. So far a characterization theorem was available for discrete Abelian groups only. Here we use a kind of localization concept for the ideals of the Fourier algebra of the underlying group. We show that localizability of ideals is equivalent to synthesizability. Based on this equivalence we show that if spectral synthesis holds on a locally compact Abelian group, then it holds on each extensions of it by a locally compact Abelian group consisting of compact elements, and also on any extension to a direct sum with a copy of the integers. Then, using Schwartz’s result and Gurevich’s counterexamples, we apply the structure theory of locally compact Abelian groups to obtain our characterization theorem. 
Székelyhidi, László. Characterisation of Locally Compact Abelian Groups Having Spectral Synthesis. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e145. doi: 10.1017/fms.2025.10039
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