Let $G$ be a locally compact group and $H$ a closed subgroup. Then $H$ is always a set of local spectral synthesis with respect to the algebra $A_p\left(G\right)$, where $A_2\left(G\right)$ is the Fourier algebra in the sense of Eymard. Global synthesis holds if and only if a certain condition (C) is satisfied; it is whenever the subgroup $H$ is amenable or normal. Global synthesis implies that each convolution operator on $L^p\left(G\right)$ with support in $H$ which is the ultraweak limit of measures carried by $H$. The problem of passing from local to global...