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.

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. ; Thangavelu, S.

Annales de l'Institut Fourier, Tome 56 (2006), p. 459-473 / Harvested from Numdam

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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.

Publié le : 2006-01-01

MR 2226023 | Zbl 1089.43006 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2189
Classification: 43A85, 53C65, 44A35

@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},
     volume = {56},
     year = {2006},
     pages = {459-473},
     doi = {10.5802/aif.2189},
     zbl = {1089.43006},
     mrnumber = {2226023},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 56 (2006) pp. 459-473. doi : 10.5802/aif.2189. http://gdmltest.u-ga.fr/item/AIF_2006__56_2_459_0/

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