On Spectral Analysis and Spectral Synthesis in the Space of Tempered Functions on Discrete Abelian Groups

被引：0
作者：
S. S. Platonov
机构：
[1] Petrozavodsk State University,Institute of Mathematics
来源：
Journal of Fourier Analysis and Applications | 2018年 / 24卷
关键词：
Spectral synthesis; Spectral analysis; Locally compact Abelian groups; Tempered functions; Bruhat–Schwartz functions; 43A45; 43A25;
D O I：
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摘要：
We consider some problems of spectral analysis and spectral synthesis in the topological vector space M(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {M}}}(G)$$\end{document} of tempered functions on a discrete Abelian group G. It is proved that spectral analysis holds in the space M(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {M}}}(G)$$\end{document} on every Abelian group G, that is, every nonzero closed linear translation invariant subspace of M(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {M}}}(G)$$\end{document} contains an exponential. For any finitely generated Abelian group G it is proved, that spectral synthesis holds in M(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {M}}}(G)$$\end{document}, that is, every closed linear translation invariant subspace H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathscr {H}}}$$\end{document} of M(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {M}}}(G)$$\end{document} coincides with the closed linear span of all exponential monomials belonging to H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathscr {H}}}$$\end{document}. For any Abelian group G with infinite torsion free rank it is proved that spectral synthesis fails to hold in the space M(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {M}}}(G)$$\end{document}.
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页码：1340 / 1376
页数：36
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