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Mat. Sb., 2013, Volume 204, Number 9, Pages 99–114 (Mi msb8173)  

This article is cited in 5 scientific papers (total in 5 papers)

On spectral synthesis on zero-dimensional Abelian groups

S. S. Platonov

Petrozavodsk State University

Abstract: 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, Fourier transform on groups.

DOI: https://doi.org/10.4213/sm8173

Full text: PDF file (535 kB)
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English version:
Sbornik: Mathematics, 2013, 204:9, 1332–1346

Bibliographic databases:

UDC: 517.986.62
MSC: Primary 43A45; Secondary 43A40
Received: 01.09.2012 and 13.03.2013

Citation: S. S. Platonov, “On spectral synthesis on zero-dimensional Abelian groups”, Mat. Sb., 204:9 (2013), 99–114; Sb. Math., 204:9 (2013), 1332–1346

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. S. Platonov, “On spectral synthesis on element-wise compact Abelian groups”, Sb. Math., 206:8 (2015), 1150–1172  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. L. Székelyhidi, “On the powers of maximal ideals in the measure algebra”, Banach J. Math. Anal., 10:2 (2016), 385–399  crossref  mathscinet  zmath  isi  scopus
    3. 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  mathnet  crossref
    4. 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  crossref  mathscinet  zmath  isi  scopus
    5. 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  crossref  mathscinet  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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