A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on $\mathbb{C}^n$

Narayanan, E. K.; Thangavelu, S.

doi:10.5802/aif.2189

A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on

ℂ^{n}

[Un théorème de Paley-Wiener spectral pour le groupe d’Heisenberg et un théorème de support pour les moyennes shériques tordues sur

ℂ^{n}

]

Narayanan, E. K.¹ ; Thangavelu, S.

¹ Indian Institute of Science Department of Mathematics Bangalore 560 012 (India)

Annales de l'Institut Fourier, Tome 56 (2006) no. 2, pp. 459-473.

Résumé
Abstract

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 $f (z) e^{\frac{1}{4} {| z |}^{2}}$ 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 $f \times μ_{r} (z) = 0$ 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 $f \times μ_{r} (z) = μ_{r} \times f (z) = 0$ 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 $ℂ^{n} .$ If $f (z) e^{\frac{1}{4} {| z |}^{2}}$ is a Schwartz class function we show that $f$ is supported in a ball of radius $B$ in $ℂ^{n}$ if and only if $f \times μ_{r} (z) = 0$ 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 $f \times μ_{r} (z) = μ_{r} \times f (z) = 0$ 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.

MR Zbl | 1 citation dans Numdam

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

Affiliations des auteurs :

Narayanan, E. K. ¹ ; Thangavelu, S.

¹ Indian Institute of Science Department of Mathematics Bangalore 560 012 (India)

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     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},
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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://www.numdam.org/articles/10.5802/aif.2189/

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