Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Beurling's theorem for Riemannian symmetric spaces II

Authors: Rudra P. Sarkar and Jyoti Sengupta
Journal: Proc. Amer. Math. Soc. 136 (2008), 1841-1853
MSC (2000): Primary 22E30, 43A85
Published electronically: December 5, 2007
MathSciNet review: 2373616
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove two versions of Beurling's theorem for Riemannian symmetric spaces of arbitrary rank. One of them uses the group Fourier transform and the other uses the Helgason Fourier transform. This is the master theorem in the quantitative uncertainty principle.

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

  • 1. Anker J. -P. A basic inequality for scattering theory on Riemannian symmetric spaces of the noncompact type, Amer. Jr. Math. 113 (1991), no. 3, 391-398. MR1109344 (92k:43008) MR 1109344 (92k:43008)
  • 2. Bonami,  A; Demange, B.; Jaming, P. Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms, Rev. Mat. Iberoamericana 19 (2003), no. 1, 23-55. MR1993414 (2004f:42015) MR 1993414 (2004f:42015)
  • 3. Cowling, M.; Sitaram, A.; Sundari, M. Hardy's uncertainty principle on semisimple groups. Pacific J. Math. 192 (2000), no. 2, 293-296. MR1744570 (2001c:22007) MR 1744570 (2001c:22007)
  • 4. Eguchi, M. Some properties of Fourier transform on Riemannian symmetric spaces. Lectures on harmonic analysis on Lie groups and related topics, pp. 9-43, Lectures in Math., 14, Kinokuniya Book Store, Tokyo, 1982. MR0683464 (84h:43026) MR 683464 (84h:43026)
  • 5. Folland, G. B. Introduction to partial differential equations. Princeton University Press, Princeton, NJ, 1995. MR1357411 (96h:35001) MR 1357411 (96h:35001)
  • 6. Folland, G. B.; Sitaram, A. The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3 (1997), no. 3, 207-238. MR1448337 (98f:42006) MR 1448337 (98f:42006)
  • 7. Gangolli, R., Varadarajan, V. S. Harmonic analysis of spherical functions on real reductive groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, 101. Springer-Verlag, Berlin, 1988. MR0954385 (89m:22015) MR 954385 (89m:22015)
  • 8. Havin, V., Jöricke, B. The uncertainty principle in harmonic analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 28. Springer-Verlag, Berlin, 1994. MR1303780 (96c:42001) MR 1303780 (96c:42001)
  • 9. Helgason, S. The Abel, Fourier and Radon Transforms on Symmetric Spaces. Indagationes Mathematicae 16 (2005), 531-551. MR 2313637
  • 10. Helgason S. Geometric Analysis on Symmetric Spaces, Mathematical Surveys and Monographs, 39. American Mathematical Society, Providence, RI, 1994 MR 96h:43009 MR 1280714 (96h:43009)
  • 11. Hörmander, L. A uniqueness theorem of Beurling for Fourier transform pairs. Ark. Mat. 29 (1991), no. 2, 237-240. MR1150375 (93b:42016) MR 1150375 (93b:42016)
  • 12. Narayanan, E. K.; Ray, S. K. $ L\sp p$ version of Hardy's theorem on semisimple Lie groups. Proc. Amer. Math. Soc. 130 (2002), no. 6, 1859-1866 MR1887035 (2003a:22009) MR 1887035 (2003a:22009)
  • 13. Sarkar, R. P.; Sengupta, J. Beurling's Theorem for Riemannian Symmetric Spaces of noncompact type preprint 2005.
  • 14. Sengupta, J. The uncertainty principle on Riemannian symmetric spaces of the noncompact type. Proc. Amer. Math. Soc. 130 (2002), no. 4, 1009-1017 MR 1873774 (2003a:43009)
  • 15. Sitaram, A.; Sundari, M. 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. MR1873774 (2003a:43009) MR 1444779 (99a:22018)
  • 16. Thangavelu, S. An introduction to the uncertainty principle. Hardy's theorem on Lie groups. Progress in Mathematics, 217. Birkhäuser Boston, Inc., Boston, MA, 2004. MR2008480 (2004j:43007) MR 2008480 (2004j:43007)

Similar Articles

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

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

Additional Information

Rudra P. Sarkar
Affiliation: Stat-Math Unit, Indian Statistical Institute, 203 B. T. Rd., Calcutta 700108, India

Jyoti Sengupta
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Rd., Mumbai 400005, India

Keywords: Beurling's theorem, uncertainty principle, symmetric space, Radon transform
Received by editor(s): November 9, 2006
Published electronically: December 5, 2007
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society