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Bent Ørsted

Generalized Fourier transforms Fk,a

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  • Ben Said Salem, Denmark
  • Toshiyuki Kobayashi, University of Tokyo, Japan
  • Bent Ørsted
  • Department of Mathematical Sciences
We construct a two-parameter family of actions ωk,a of the Lie algebra View the MathML source by differential-difference operators on View the MathML source. 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 View the MathML source, 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 transforms View the MathML source, which includes the Dunkl transform View the MathML source (a=2) and a new unitary operator View the MathML source (a=1) as a Dunkl-type generalization of the classical Hankel transform.
Translated title of the contributionTransformation de Fourier généralisée Fk,a
Original languageEnglish
JournalComptes Rendus Mathématique
Pages (from-to)1119-1124
Number of pages6
Publication statusPublished - 2009

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