TY - JOUR

T1 - Generalized Fourier transforms Fk, a

AU - Ben Saïd, Salem

AU - Kobayashi, Toshiyuki

AU - Ørsted, Bent

N1 - Funding Information:
E-mail addresses: Salem.BenSaid@iecn.u-nancy.fr (S. Ben Saïd), toshi@ms.u-tokyo.ac.jp (T. Kobayashi), orsted@imf.au.dk (B. Ørsted). 1 Partially supported by Grant-in-Aid for Scientific Research (B) (18340037), Japan Society for the Promotion of Science, Max Planck Institute at Bonn, and the Alexander Humboldt Foundation.

PY - 2009/10

Y1 - 2009/10

N2 - 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 (k ≡ 0, a = 2) and the Laguerre semigroup by T. Kobayashi and G. Mano (k ≡ 0, a = 1). The boundary value of our semigroup Ωk, a provides us with (k, a)-generalized Fourier transformsFk, 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. To cite this article: S. Ben Saïd et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

AB - 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 (k ≡ 0, a = 2) and the Laguerre semigroup by T. Kobayashi and G. Mano (k ≡ 0, a = 1). The boundary value of our semigroup Ωk, a provides us with (k, a)-generalized Fourier transformsFk, 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. To cite this article: S. Ben Saïd et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

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U2 - 10.1016/j.crma.2009.07.015

DO - 10.1016/j.crma.2009.07.015

M3 - Article

AN - SCOPUS:71749103626

SN - 1631-073X

VL - 347

SP - 1119

EP - 1124

JO - Comptes Rendus Mathematique

JF - Comptes Rendus Mathematique

IS - 19-20

ER -