Exercise 16, § 3

Let $f\in L^{\infty}(G)$. We say that $f$ satisfies the spectral synthesis condition if it belongs to $Y(A(W_f))$, where $W_f$ is the intersection of all weakly closed translation invariant subspaces of $L^{\infty}(G)$ which contain $f$.

a) Let $\mu\in \mathscr{M}^1(\widehat{G})$ and $f=\overline{\mathscr{F}}_G(\mu)\in L^{\infty}(G)$. Then $A(W_f)$ is the support of $\mu$. Moreover, $f$ satisfies the condi$\widehat{ti}$on of spectral synthesis.

b) If G is discrete, every function $f\in L^2(G)$ satisfies the spectral synthesis condition.