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 Mat. Sb., 2015, Volume 206, Number 8, Pages 127–152 (Mi msb8419)

On spectral synthesis on element-wise compact Abelian groups

S. S. Platonov

Petrozavodsk State University

Abstract: Let $G$ be an arbitrary locally compact Abelian group and let $C(G)$ be the space of all continuous complex-valued functions on $G$. A closed linear subspace $\mathscr H\subseteq C(G)$ is referred to as an invariant subspace if it is invariant with respect to the shifts $\tau_y\colon f(x)\mapsto f(xy)$, $y\in G$. By definition, an invariant subspace $\mathscr H\subseteq C(G)$ admits strict spectral synthesis if $\mathscr H$ coincides with the closure in $C(G)$ of the linear span of all characters of $G$ belonging to $\mathscr H$. We say that strict spectral synthesis holds in the space $C(G)$ on $G$ if every invariant subspace $\mathscr H\subseteq C(G)$ admits strict spectral synthesis. An element $x$ of a topological group $G$ is said to be compact if $x$ is contained in some compact subgroup of $G$. A group $G$ is said to be element-wise compact if all elements of $G$ are compact. The main result of the paper is the proof of the fact that strict spectral synthesis holds in $C(G)$ for a locally compact Abelian group $G$ if and only if $G$ is element-wise compact.
Bibliography: 14 titles.

Keywords: spectral synthesis, locally compact Abelian groups, element-wise compact groups, Fourier transform on groups, Bruhat-Schwartz functions.

DOI: https://doi.org/10.4213/sm8419

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English version:
Sbornik: Mathematics, 2015, 206:8, 1150–1172

Bibliographic databases:

UDC: 517.986.62
MSC: 43A25

Citation: S. S. Platonov, “On spectral synthesis on element-wise compact Abelian groups”, Mat. Sb., 206:8 (2015), 127–152; Sb. Math., 206:8 (2015), 1150–1172

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8419
• https://doi.org/10.4213/sm8419
• http://mi.mathnet.ru/eng/msb/v206/i8/p127

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This publication is cited in the following articles:
1. S. S. Platonov, “O spektralnom sinteze v prostranstve funktsii medlennogo rosta na konechno porozhdennykh abelevykh gruppakh”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2016, no. 5(36), 42–59
2. S. S. Platonov, “On spectral analysis and spectral synthesis in the space of tempered functions on discrete abelian groups”, J. Fourier Anal. Appl., 24:5 (2018), 1340–1376
3. Platonov S.S., “Spectral Synthesis For the Space of Tempered Solutions of a Convolution System on Discrete Abelian Groups”, Acta Math. Hung., 157:1 (2019), 141–153
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