Generalized Fourier transforms Fk,a

Ben Said Salem; Toshiyuki Kobayashi; Bent Ørsted

doi:10.1016/j.crma.2009.07.015

Generalized Fourier transforms F_k,a

Bidragets oversatte titel: Transformation de Fourier généralisée F_k,a

Ben Said Salem, Toshiyuki Kobayashi, Bent Ørsted

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

Abstract

We construct a two-parameter family of actions ω_k,a of the Lie algebra

by differential-difference operators on

. 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 ω_k,a lifts to a unitary representation of the universal covering of

, and can even be extended to a holomorphic semigroup Ω_k,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 Ω_k,a provides us with (k,a)-generalized Fourier transforms

, which includes the Dunkl transform

(a=2) and a new unitary operator

(a=1) as a Dunkl-type generalization of the classical Hankel transform.

Bidragets oversatte titel	Transformation de Fourier généralisée F_k,a
Originalsprog	Engelsk
Tidsskrift	Comptes Rendus Mathématique
Vol/bind	347
Nummer	19-20
Sider (fra-til)	1119-1124
Antal sider	6
ISSN	1631-073X
DOI	https://doi.org/10.1016/j.crma.2009.07.015
Status	Udgivet - 2009

Adgang til dokumentet

10.1016/j.crma.2009.07.015

Citationsformater

@article{a988be60af4c11dea554000ea68e967b,

title = "Generalized Fourier transforms Fk,a",

abstract = "We construct a two-parameter family of actions ωk,a of the Lie algebra by differential-difference operators on . 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 , 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 , which includes the Dunkl transform (a=2) and a new unitary operator (a=1) as a Dunkl-type generalization of the classical Hankel transform.",

author = "Salem, {Ben Said} and Toshiyuki Kobayashi and Bent {\O}rsted",

year = "2009",

doi = "10.1016/j.crma.2009.07.015",

language = "English",

volume = "347",

pages = "1119--1124",

journal = "Comptes Rendus Math{\'e}matique",

issn = "1631-073X",

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TY - JOUR

T1 - Generalized Fourier transforms Fk,a

AU - Salem, Ben Said

AU - Kobayashi, Toshiyuki

AU - Ørsted, Bent

PY - 2009

Y1 - 2009

N2 - We construct a two-parameter family of actions ωk,a of the Lie algebra by differential-difference operators on . 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 , 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 , which includes the Dunkl transform (a=2) and a new unitary operator (a=1) as a Dunkl-type generalization of the classical Hankel transform.

AB - We construct a two-parameter family of actions ωk,a of the Lie algebra by differential-difference operators on . 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 , 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 , which includes the Dunkl transform (a=2) and a new unitary operator (a=1) as a Dunkl-type generalization of the classical Hankel transform.

U2 - 10.1016/j.crma.2009.07.015

DO - 10.1016/j.crma.2009.07.015

M3 - Journal article

SN - 1631-073X

VL - 347

SP - 1119

EP - 1124

JO - Comptes Rendus Mathématique

JF - Comptes Rendus Mathématique

IS - 19-20

ER -

Generalized Fourier transforms Fk,a

Abstract

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Generalized Fourier transforms F_k,a