A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on n
[Un théorème de Paley-Wiener spectral pour le groupe d’Heisenberg et un théorème de support pour les moyennes shériques tordues sur n ]
Annales de l'Institut Fourier, Tome 56 (2006) no. 2, pp. 459-473.

Nous prouvons un théorème de Paley-Wiener spectral pour le groupe d’Heisenberg en utilisant un théorème du support pour les moyennes sphériques tordues sur n . Si f(z)e 1 4|z| 2 est une fonction dans la classe de Schwartz nous montrons que f a un support dans une boule de n de rayon B si et seulement si f×μ r (z)=0 pour r>B+|z| et pour tout z n . C’est un analogue du théorème du support prouvé dans les contextes euclidiens et hyperboliques par Helgason. Lorsque n=1 nous montrons que les deux conditions f×μ r (z)=μ r ×f(z)=0 pour r>B+|z| impliquent un théorème du support pour une grande classe de fonctions à croissance exponentielle. Il est surprenant de constater que ce dernier résultat ne se généralise pas aux dimensions supérieures.

We prove a spectral Paley-Wiener theorem for the Heisenberg group by means of a support theorem for the twisted spherical means on n . If f(z)e 1 4|z| 2 is a Schwartz class function we show that f is supported in a ball of radius B in n if and only if f×μ r (z)=0 for r>B+|z| for all z n . This is an analogue of Helgason’s support theorem on Euclidean and hyperbolic spaces. When n=1 we show that the two conditions f×μ r (z)=μ r ×f(z)=0 for r>B+|z| imply a support theorem for a large class of functions with exponential growth. Surprisingly enough,this latter result does not generalize to higher dimensions.

DOI : 10.5802/aif.2189
Classification : 43A85, 53C65, 44A35
Mots clés : Spectral Paley-Wiener theorem, twisted spherical means, special Hermite operator, Laguerre functions, support theorem, spherical harmonics
Narayanan, E. K. 1 ; Thangavelu, S. 

1 Indian Institute of Science Department of Mathematics Bangalore 560 012 (India)
@article{AIF_2006__56_2_459_0,
     author = {Narayanan, E.~K. and Thangavelu, S.},
     title = {A spectral {Paley-Wiener} theorem for the {Heisenberg} group and a support theorem for the twisted spherical means on $\mathbb{C}^n$},
     journal = {Annales de l'Institut Fourier},
     pages = {459--473},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
     number = {2},
     year = {2006},
     doi = {10.5802/aif.2189},
     zbl = {1089.43006},
     mrnumber = {2226023},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2189/}
}
TY  - JOUR
AU  - Narayanan, E. K.
AU  - Thangavelu, S.
TI  - A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on $\mathbb{C}^n$
JO  - Annales de l'Institut Fourier
PY  - 2006
SP  - 459
EP  - 473
VL  - 56
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2189/
DO  - 10.5802/aif.2189
LA  - en
ID  - AIF_2006__56_2_459_0
ER  - 
%0 Journal Article
%A Narayanan, E. K.
%A Thangavelu, S.
%T A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on $\mathbb{C}^n$
%J Annales de l'Institut Fourier
%D 2006
%P 459-473
%V 56
%N 2
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2189/
%R 10.5802/aif.2189
%G en
%F AIF_2006__56_2_459_0
Narayanan, E. K.; Thangavelu, S. A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on $\mathbb{C}^n$. Annales de l'Institut Fourier, Tome 56 (2006) no. 2, pp. 459-473. doi : 10.5802/aif.2189. http://www.numdam.org/articles/10.5802/aif.2189/

[1] Agranovsky, M. L.; Rawat, Rama Injectivity sets for the spherical means on the Heisenberg group, J. Fourier Anal. Appl., Volume 5 (1999) no. 4, pp. 363-372 | DOI | MR | Zbl

[2] Bray, W. O. A spectral Paley-Wiener theorem, Monatsh. Math., Volume 116 (1993) no. 1, pp. 1-11 | DOI | MR | Zbl

[3] Bray, W. O. Generalized spectral projections on symmetric spaces of non compact type: Paley-Wiener theorems, J. Funct. Anal., Volume 135 (1996) no. 1, pp. 206-232 | DOI | MR | Zbl

[4] Epstein, C. L.; Kleiner, B. Spherical means in annular regions, Comm. Pure Appl. Math., Volume 46 (1993) no. 3, pp. 441-451 | DOI | MR | Zbl

[5] Helgason, S. Groups and Geometric Analysis, Academic press, New York, 1984 | MR | Zbl

[6] Narayanan, E. K.; Thangavelu, S. Injectivity sets for the spherical means on the Heisenberg group, J. Math. Anal. Appl., Volume 263 (2001) no. 2, pp. 565-579 | DOI | MR | Zbl

[7] Olver, F. W. J. Asymptotics and special functions, Academic press, New York, 1974 (Computer Science and Applied Mathematics) | MR | Zbl

[8] Rudin, W. Function theory in the unit ball of n , 241, Springer-Verlag, New York - Berlin, 1980 | MR | Zbl

[9] Sajith, G.; Thangavelu, S. On the injectivity of twisted spherical means on n , Israel J. Math., Volume 122 (2), pp. 79-92 | DOI | MR | Zbl

[10] Strichartz, R. S. Harmonic analysis as spectral theory of Laplacians, J. Funct. Anal., Volume 87 (1989), pp. 51-148 | DOI | MR | Zbl

[11] Szego, G. Orthogonal polynomials, Colloq. Pub., 23, Amer. Math. Soc., Providence, R. I., 1967

[12] Thangavelu, S. Lectures on Hermite and Laguerre expansions, Mathematical Notes, 42, Princeton University Press, Princeton, NJ, 1993 | MR | Zbl

[13] Thangavelu, S. Harmonic analysis on the Heisenberg group, Progress in Mathematics, 159, Birkhäuser Boston, Boston, MA, 1998 | MR | Zbl

[14] Thangavelu, S. An introduction to the uncertainty principle, Progress in Mathematics, 217, Birkhäuser Boston, Boston, MA, 2004 | MR | Zbl

Cité par Sources :