Geodesic
Spectral synthesis on zero-dimensional locally compact abelian groups
Vestnik rossijskih universitetov. Matematika, Tome 24 (2019) no. 128, pp. 450-456

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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 ${\mathcal H}.$
Keywords: zero-dimensional groups, characters, harmonic analysis, spectral synthesis, invariant subspaces. 
@article{VTAMU_2019_24_128_a7,
     author = {S. S. Platonov},
     title = {Spectral synthesis on zero-dimensional locally compact abelian groups},
     journal = {Vestnik rossijskih universitetov. Matematika},
     pages = {450--456},
     publisher = {mathdoc},
     volume = {24},
     number = {128},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTAMU_2019_24_128_a7/}
}
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S. S. Platonov. Spectral synthesis on zero-dimensional locally compact abelian groups. Vestnik rossijskih universitetov. Matematika, Tome 24 (2019) no. 128, pp. 450-456. http://geodesic.mathdoc.fr/item/VTAMU_2019_24_128_a7/