Spectral synthesis

Remarks

Every closed ideal $I$ with hull $E$ satisfies $I \subseteq I(E)$, so $E$ is a set of synthesis exactly when $I(E)$ is the smallest closed ideal with hull $E$. The hull of $I(E)$ is $E$ because $L^1(G)$ is a regular algebra: for $\chi \notin E$ there is $f$ whose transform vanishes on $E$ but not at $\chi$.

Example: $E = \emptyset$ is a set of synthesis. Here $I(\emptyset) = L^1(G)$, and Wiener's Tauberian theorem says that a closed ideal whose transforms have no common zero is all of $L^1(G)$.

Example: if $G$ is compact, then $\widehat{G}$ is discrete and spectral synthesis holds for $G$.

Non-example: for $G = \R^3$, the unit sphere in $\widehat{\R^3} \cong \R^3$ is not a set of spectral synthesis, by a result of Laurent Schwartz.

Other names are a set of synthesis and a set that admits spectral synthesis.