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Scientific articles

Spectral synthesis on zero-dimensional locally compact abelian groups

S. S. Platonov

Petrozavodsk State University

Abstract: 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.

UDC: 517.986.62

Received: 19.08.2019

DOI: 10.20310/2686-9667-2019-24-128-450-456