Spectral Synthesis on Varieties

Abstract

In his classical paper, Laurent Schwartz proved that on the real line, in every linear translation invariant space of continuous complex valued functions, which is closed under compact convergence the exponential monomials span a dense subspace. He studied so-called local ideals in the space of Fourier transforms, and his proof based on the observation that, on the one hand, these local ideals are completely determined by the exponential monomials in the space, and, on the other hand, these local ideals completely determine the space itself. On the other hand, Dimitri Gurevich gave counterexamples for Schwartz's theorem in higher dimension. In this paper we show that the ideas of localisation can be extended to general locally compact Abelian groups using abstract derivations on the Fourier algebra of compactly supported measures. Based on this method we present necessary and sufficient conditions for spectral synthesis for varieties on locally compact Abelian groups. Using localisation, in \cite{MR4789359} we proved that spectral synthesis holds on a locally compact Abelian group $G$ if and only if it holds on $G/B$, where $B$ is the closed subgroup of compact elements. This may lead to a complete characterisation of locally compact Abelian groups having spectral synthesis.