Comptes Rendus
Lie Algebras/Harmonic Analysis
Generalized Fourier transforms Fk,a
Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1119-1124.

We construct a two-parameter family of actions ωk,a of the Lie algebra sl(2,R) by differential-difference operators on RN. Here, k is a multiplicity-function for the Dunkl operators, and a>0 arises from the interpolation of the Weil representation and the minimal unitary representation of the conformal group. The action ωk,a lifts to a unitary representation of the universal covering of SL(2,R), and can even be extended to a holomorphic semigroup Ωk,a. Our semigroup generalizes the Hermite semigroup studied by R. Howe (k0, a=2) and the Laguerre semigroup by T. Kobayashi and G. Mano (k0, a=1). The boundary value of our semigroup Ωk,a provides us with (k,a)-generalized Fourier transforms Fk,a, which includes the Dunkl transform Dk (a=2) and a new unitary operator Hk (a=1) as a Dunkl-type generalization of the classical Hankel transform.

À l'aide des opérateurs différentiels et aux différences de Dunkl sur RN, on construit une famille d'actions ωk,a de l'algèbre de Lie sl(2,R) dépendant de deux paramètres k et a. Ici k est une fonction de multiplicité associée aux opérateurs de Dunkl, et a>0 un paramètre d'interpolation entre la représentation de Weil et la représentation minimale du groupe conforme. On montre que ωk,a s'intègre à une représentation unitaire du revêtement universel du groupe SL(2,R), et se prolonge à un semi-groupe holomorphe Ωk,a. Notre semi-groupe généralise le semi-groupe de Hermite, étudié par R. Howe (k0, a=2), ainsi que le semi-groupe de Laguerre dû à T. Kobayashi et G. Mano (k0, a=1). La valeur au bord de notre semi-groupe Ωk,a donne une transformation de Fourier (k,a)-généralisée Fk,a qui correspond à la transformation de Dunkl pour a=2, et à une nouvelle transformation Hk pour a=1 qui généralise la transformation de Hankel classique.

Published online:
DOI: 10.1016/j.crma.2009.07.015

Salem Ben Saïd 1; Toshiyuki Kobayashi 2; Bent Ørsted 3

1 Université Henri-Poincaré-Nancy 1, Institut Elie-Cartan, B.P. 239, 54506 Vandoeuvre-Les-Nancy, France
2 The University of Tokyo, Graduate School of Mathematical Sciences, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan
3 University of Aarhus, Department of Mathematical Sciences, Ny Munkegade, DK 8000, Aarhus C, Denmark
     author = {Salem Ben Sa{\"\i}d and Toshiyuki Kobayashi and Bent {\O}rsted},
     title = {Generalized {Fourier} transforms $ {\mathcal{F}}_{k,a}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1119--1124},
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     year = {2009},
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Salem Ben Saïd; Toshiyuki Kobayashi; Bent Ørsted. Generalized Fourier transforms $ {\mathcal{F}}_{k,a}$. Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1119-1124. doi : 10.1016/j.crma.2009.07.015.

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