Spectral synthesis

Summary

Let $G$ be a locally compact abelian group with group algebra $L^1(G)$. Spectral synthesis asks whether a closed ideal of $L^1(G)$ is determined by the characters where all its Fourier transforms vanish. A closed set $E$ of the dual group is a set of spectral synthesis if exactly one closed ideal has $E$ as zero set. Wiener's Tauberian theorem is the case $E = \emptyset$, and points are also sets of synthesis. By Malliavin's theorem, synthesis fails for some closed set as soon as the dual group is not discrete.