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Harmonic analysis on discrete Abelian groups

Authors: M. Laczkovich and G. Székelyhidi
Journal: Proc. Amer. Math. Soc. 133 (2005), 1581-1586
MSC (2000): Primary 20K99; Secondary 43A45, 12F05
Published electronically: December 6, 2004
MathSciNet review: 2120269
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be an Abelian group and let $\mathbb C^G$ denote the linear space of all complex-valued functions defined on $G$ equipped with the product topology. We prove that the following are equivalent.

(i) Every nonzero translation invariant closed subspace of $\mathbb C^G$contains an exponential; that is, a nonzero multiplicative function.

(ii) The torsion free rank of $G$ is less than the continuum.

References [Enhancements On Off] (What's this?)

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  • 2. R. J. Elliot, Two notes on spectral synthesis for discrete Abelian groups, Proc. Cambridge Phil. Soc. 61 (1965), 617-620. MR 0177260 (31:1523)
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Additional Information

M. Laczkovich
Affiliation: Department of Analysis, Eötvös Loránd University, Budapest, Pázmány Péter sétány 1/C, 1117 Hungary – and – Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, England

G. Székelyhidi
Affiliation: Department of Mathematics, Imperial College, Huxley Building, 180 Queen’s Gate, London, SW7 2AZ, England

Keywords: Problem of harmonic analysis, exponential functions, Hilbert's Nullstellensatz
Received by editor(s): February 10, 2004
Published electronically: December 6, 2004
Additional Notes: The research of the first author was partially supported by the Hungarian National Foundation for Scientific Research, Grant No. T032042
Communicated by: Andreas Seeger
Article copyright: © Copyright 2004 American Mathematical Society

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