Discrete spectral synthesis

Discrete spectral analysis and synthesis study the description of translation invariant function spaces over discrete Abelian groups. The basic building bricks are the exponential monomials. A remarkable result of R. J. Elliot in 1965 claimed that spectral synthesis holds on any Abelian group, which means that the exponential monomials span a dense linear subspace in any pointwise-closed translation invariant linear space of complex valued functions over the group. Unfortunately, the proof of this theorem had several gaps. In this paper we give a short survey about the present status of discrete spectral analysis and synthesis, we show that Elliot’s theorem is false, we give a necessary condition for Abelian groups to have spectral synthesis and we formulate a conjecture about a possible characterization of Abelian groups having spectral synthesis.