Spectral synthesis on zero-dimensional locally compact abelian groups

Let $G$ be a zero-dimensional locally compact Abelian group whose elements are compact, $C(G)$ the space of continuous complex-valued functions on the group $G$. A closed linear subspace ${\mathcal H}\subseteq C(G)$ is called invariant subspace, if it is invariant with respect to translations $\tau_y: f(x)\mapsto f(x+y)$, $y\in G$. We prove that any invariant subspace ${\mathcal H}$ admits spectral synthesis, which means that ${\mathcal H}$ coincides with the closure of the linear span of all characters of the group $G$ contained in ${\mathcal H}.$