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In this paper we solve a long-standing problem which goes back to LaurentSchwartz's work on mean periodic functions. Namely, we completely characterisethose locally compact Abelian groups having spectral synthesis. So far acharacterisation theorem was available for discrete Abelian groups only. Herewe use a kind of localisation concept for the ideals of the Fourier algebra ofthe underlying group. We show that localisability of ideals is equivalent tosynthesizability. Based on this equivalence we show that if spectral synthesisholds on each extension of it by a locally compact Abelian group consisting ofcompact elements, and also on any extension to a direct sum with a copy of theintegers. Then, using Schwartz's result and Gurevich's counterexamples we applythe structure theory of locally compact Abelian groups.

Characterisation of Locally Compact Abelian Groups Having Spectral Synthesis