A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on n
Annales de l'Institut Fourier, Volume 56 (2006) no. 2, p. 459-473

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.

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.

DOI : https://doi.org/10.5802/aif.2189
Classification:  43A85,  53C65,  44A35
Keywords: Spectral Paley-Wiener theorem, twisted spherical means, special Hermite operator, Laguerre functions, support theorem, spherical harmonics
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {2},
     year = {2006},
     pages = {459-473},
     doi = {10.5802/aif.2189},
     mrnumber = {2226023},
     zbl = {1089.43006},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 56 (2006) no. 2, pp. 459-473. doi : 10.5802/aif.2189. http://www.numdam.org/item/AIF_2006__56_2_459_0/

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