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)
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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://archive.numdam.org/articles/10.5802/aif.2189/

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