Geodesic
On spectral synthesis on zero-dimensional Abelian groups
Sbornik. Mathematics, Tome 204 (2013) no. 9, pp. 1332-1346

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Let $G$ be a zero-dimensional locally compact Abelian group all of whose elements are compact, and let $C(G)$ be the space of all complex-valued continuous functions on $G$. A closed linear subspace $\mathscr H\subseteq C(G)$ is said to be an invariant subspace if it is invariant with respect to the translations $\tau_y\colon f(x)\mapsto f(x+y)$, $y\in G$. In the paper, it is proved that any invariant subspace $\mathscr H$ admits spectral synthesis, that is, $\mathscr H$ coincides with the closed linear span of the characters of $G$ belonging to $\mathscr H$. Bibliography: 25 titles.
Keywords: spectral synthesis, locally compact Abelian group, zero-dimensional group, invariant subspace
Mots-clés : Fourier transform on groups. 
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     author = {S. S. Platonov},
     title = {On spectral synthesis on zero-dimensional {Abelian} groups},
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S. S. Platonov. On spectral synthesis on zero-dimensional Abelian groups. Sbornik. Mathematics, Tome 204 (2013) no. 9, pp. 1332-1346. http://geodesic.mathdoc.fr/item/SM_2013_204_9_a4/