# ANNALES DE L'INSTITUT FOURIER

A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on ${ℂ}^{n}$
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 ${ℂ}^{n}.$ If $f\left(z\right){e}^{\frac{1}{4}{|z|}^{2}}$ is a Schwartz class function we show that $f$ is supported in a ball of radius $B$ in ${ℂ}^{n}$ if and only if $f×{\mu }_{r}\left(z\right)=0$ for $r>B+|z|$ for all $z\in {ℂ}^{n}.$ This is an analogue of Helgason’s support theorem on Euclidean and hyperbolic spaces. When $n=1$ we show that the two conditions $f×{\mu }_{r}\left(z\right)={\mu }_{r}×f\left(z\right)=0$ 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 ${ℂ}^{n}.$ Si $f\left(z\right){e}^{\frac{1}{4}{|z|}^{2}}$ est une fonction dans la classe de Schwartz nous montrons que $f$ a un support dans une boule de ${ℂ}^{n}$ de rayon $B$ si et seulement si $f×{\mu }_{r}\left(z\right)=0$ pour $r>B+|z|$ et pour tout $z\in {ℂ}^{n}.$ 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 $f×{\mu }_{r}\left(z\right)={\mu }_{r}×f\left(z\right)=0$ 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},
zbl = {1089.43006},
mrnumber = {2226023},
language = {en},
url = {https://aif.centre-mersenne.org/item/AIF_2006__56_2_459_0}
}

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. https://aif.centre-mersenne.org/item/AIF_2006__56_2_459_0/

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