The uncertainty principle on Riemannian symmetric spaces of the noncompact type

Author:
J. Sengupta

Journal:
Proc. Amer. Math. Soc. **130** (2002), 1009-1017

MSC (2000):
Primary 43A85, 22E30

Published electronically:
August 29, 2001

MathSciNet review:
1873774

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

The uncertainty principle in says that it is impossible for a function and its Fourier transform to be simultaneously very rapidly decreasing. A quantitative assertion of this principle is Hardy's theorem. In this article we prove various generalisations of Hardy's theorem for Riemannian symmetric spaces of the noncompact type. In the case of the real line these results were obtained by Morgan and Cowling-Price.

**1.**Jean-Philippe Anker,*A basic inequality for scattering theory on Riemannian symmetric spaces of the noncompact type*, Amer. J. Math.**113**(1991), no. 3, 391–398. MR**1109344**, 10.2307/2374832**2.**S. C. Bagchi and Swagato K. Ray,*Uncertainty principles like Hardy’s theorem on some Lie groups*, J. Austral. Math. Soc. Ser. A**65**(1998), no. 3, 289–302. MR**1660417****3.**H. Dym and H. P. McKean,*Fourier series and integrals*, Academic Press, New York-London, 1972. Probability and Mathematical Statistics, No. 14. MR**0442564****4.**Ramesh Gangolli and V. S. Varadarajan,*Harmonic analysis of spherical functions on real reductive groups*, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 101, Springer-Verlag, Berlin, 1988. MR**954385****5.**Sigurdur Helgason,*Geometric analysis on symmetric spaces*, Mathematical Surveys and Monographs, vol. 39, American Mathematical Society, Providence, RI, 1994. MR**1280714****6.**Narayanan and Ray,*versions of Hardy's theorem on semi-simple groups*, Proc. AMS to appear.**7.**A. Sitaram and M. Sundari,*An analogue of Hardy’s theorem for very rapidly decreasing functions on semi-simple Lie groups*, Pacific J. Math.**177**(1997), no. 1, 187–200. MR**1444779**, 10.2140/pjm.1997.177.187**8.**A. Sitaram, M. Sundari, and S. Thangavelu,*Uncertainty principles on certain Lie groups*, Proc. Indian Acad. Sci. Math. Sci.**105**(1995), no. 2, 135–151. MR**1350473**, 10.1007/BF02880360

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
43A85,
22E30

Retrieve articles in all journals with MSC (2000): 43A85, 22E30

Additional Information

**J. Sengupta**

Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay - 400 005, India

Email:
sengupta@math.tifr.res.in

DOI:
http://dx.doi.org/10.1090/S0002-9939-01-06318-3

Received by editor(s):
September 18, 2000

Published electronically:
August 29, 2001

Communicated by:
Rebecca Herb

Article copyright:
© Copyright 2001
American Mathematical Society