Spectral synthesis on the group of conformal automorphisms of the unit disc

Given a group $G$ of conformal automorphisms of the unit disc $\mathbb{D}$, the problem of spectral synthesis is solved for subspaces $\mathscr{U}\subset \mathscr{E}(G)$ that are invariant under right $G$-shifts and conjugations by elements of the subgroup $\operatorname{SO}(2)$. A new theorem on spectral synthesis for subspaces of $\mathscr{E}(\mathbb{D})$ that are invariant under weighted conformal shifts is an intermediate result.
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