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Uncertainty principles like Hardy's theorem on some Lie groups

Part of: Lie groups

Published online by Cambridge University Press:  09 April 2009

S. C. Bagchi  and
Swagato K. Ray
S. C. Bagchi
Affiliation:
Stat-Math Division, Indian Statistical Institute, 203, B. T. Road, Calcutta 700 035, India e-mail: [email protected] & [email protected]
Swagato K. Ray
Affiliation:
Stat-Math Division, Indian Statistical Institute, 203, B. T. Road, Calcutta 700 035, India e-mail: [email protected] & [email protected]
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Abstract

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We extend an uncertainty principle due to Cowling and Price to Euclidean spaces, Heisenberg groups and the Euclidean motion group of the plane. This uncertainty principle is a generalisation of a classical result due to Hardy. We also show that on the real line this uncertainty principle is almost equivalent to Hardy's theorem.

Keywords

Uncertainty principlesHeisenberg groupsEuclidean motion groupOscillator groupHardy's theorem

MSC classification

Secondary: 22E25: Nilpotent and solvable Lie groups 22E30: Analysis on real and complex Lie groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 65 , Issue 3 , December 1998 , pp. 289 - 302
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Cowling, M. G. and Price, J. F., ‘Generalizations of Heisenberg's inequality’, in: Harmonic analysis (eds. Mauceri, G., Ricci, F. and Weiss, G.), LNM, no.992 (Springer, Berlin, 1983) pp. 443–449.Google Scholar
[2]Folland, G. B., A course in abstract Harmonic Analysis (CRC Press. London, 1995).Google Scholar
[3]Folland, G. B. and Sitaram, A., ‘The uncertainty principles: A mathematical survey’, J. Fourier Anal. Appl. 3 (1997), 207–238.CrossRefGoogle Scholar
[4]Hardy, G. H., ‘A theorem concerning Fourier transform’, J. London Math. Soc. 8 (1933), 227–231.Google Scholar
[5]Havin, V. and Joricke, B., The uncertainty principle in harmonic analysis (Springer, Berlin, 1994).CrossRefGoogle Scholar
[6]Hörmander, L., ‘A uniqueness theorem of Beurling for Fourier transform pairs’, Ark. Mat. 29 (1991), 237–240.CrossRefGoogle Scholar
[7]Kleppner, A. and Lipsman, R. L., ‘The Plancherel formula for group extensions (II)’, Ann. Sci. Ecole. Norm. Sup. (4), 6 (1973), 103–132.CrossRefGoogle Scholar
[8]Morgan, G. W., ‘A note on Fourier transforms’, J. London Math. Soc. 9 (1934), 187–192.Google Scholar
[9]Parthasarathy, K. R., Multipliers on locally compact groups, LNM, no. 93 (Springer, Berlin, 1969).Google Scholar
[10]Sitaram, A. and Sundari, M., ‘An analogue of Hardy's theorem for very rapidly decreasing functions on semisimple Lie groups’, Pacific J. Math. 177 (1997), 187–200.Google Scholar
[11]Sitaram, A., Sundari, M. and Thangavelu, S., ‘Uncertainty principles on certain Lie groups’, Proc. Ind. Acad. Sci. (Math. Sci.) 105 (1995), 135–151.Google Scholar
[12]Sugiura, M., Unitary representations and Harmonic Analysis, an introduction (Kodansha Scientific books, Tokyo, 1975).Google Scholar
[13]Sundari, M., ‘Hardy's theorem for the n-dimensional Euclidean motion group’, Proc. Amer. Math. Soc. 126 (1998), 1199–1204.CrossRefGoogle Scholar