Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Beurling's theorem for Riemannian symmetric spaces II

Author(s): Rudra P. Sarkar; Jyoti Sengupta
Journal: Proc. Amer. Math. Soc. 136 (2008), 1841-1853.
MSC (2000): Primary 22E30, 43A85
Posted: December 5, 2007
MathSciNet review: 2373616
Retrieve article in: 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:

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)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|