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Harmonic analysis on discrete Abelian groups
Author(s):
M.
Laczkovich;
G.
Székelyhidi
Journal:
Proc. Amer. Math. Soc.
133
(2005),
1581-1586.
MSC (2000):
Primary 20K99;
Secondary 43A45, 12F05
Posted:
December 6, 2004
MathSciNet review:
2120269
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Abstract:
Let  be an Abelian group and let  denote the linear space of all complex-valued functions defined on  equipped with the product topology. We prove that the following are equivalent. (i) Every nonzero translation invariant closed subspace of  contains an exponential; that is, a nonzero multiplicative function. (ii) The torsion free rank of  is less than the continuum.
References:
-
- 1.
- N. Bourbaki: Commutative Algebra. Hermann and Addison-Wesley, 1972. MR 0360549 (50:12997)
- 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)
- 3.
- H. Matsumura: Commutative Ring Theory. Cambridge University Press, 1986. MR 0879273 (88h:13001)
- 4.
- G. Székelyhidi, Spectral synthesis on locally compact Abelian groups (essay). Cambridge, Trinity College,
2001. - 5.
- G. Székelyhidi, Spectral analysis, unpublished manuscript, 2002.
- 6.
- L. Székelyhidi, The failure of spectral synthesis on some types of discrete Abelian groups, J. Math. Anal. and Applications 291 (2004), no. 2, 757-763. MR 2039084
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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
Email:
laczk@cs.elte.hu
G.
Székelyhidi
Affiliation:
Department of Mathematics, Imperial College, Huxley Building, 180 Queen's Gate, London, SW7 2AZ, England
Email:
gabor.szekelyhidi@imperial.ac.uk
DOI:
10.1090/S0002-9939-04-07749-4
PII:
S 0002-9939(04)07749-4
Keywords:
Problem of harmonic analysis,
exponential functions,
Hilbert's Nullstellensatz
Received by editor(s):
February 10, 2004
Posted:
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
Copyright of article:
Copyright
2004,
American Mathematical Society
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