Generalized Fock spaces and Weyl relations for the Dunkl kernel on the real line

Sifi, Mohamed; Soltani, Fethi

doi:10.1016/s0022-247x(02)00052-5

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Introduction4

Commutators and Weyl Relations For The Dunkl Kernel1

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2004

2020

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Cited by 33 publications

(24 citation statements)

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“…2) If d = 1 and G = Z 2 [21], the Weyl commutation relations are given by E γ (bQ)E γ (aT γ ) = E γ (aT γ )E γ (bP a );…”

Section: Commutators and Weyl Relations For The Dunkl Kernelmentioning

confidence: 99%

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Generalized Fock spaces and Weyl commutation relations for the Dunkl kernel

Soltani¹

2004

Pacific J. Math.

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“…2) If d = 1 and G = Z 2 [21], the Weyl commutation relations are given by E γ (bQ)E γ (aT γ ) = E γ (aT γ )E γ (bP a );…”

Section: Commutators and Weyl Relations For The Dunkl Kernelmentioning

confidence: 99%

“…Next, the ordinary Fock space A is the subject of many works ( [5,7] and [8]). In 2001, M. Sifi and F. Soltani [21] introduced a Hilbert space A γ of entire functions where the inner product is weighted by a generalized Gaussian distribution. On A γ the Dunkl operator on the real line:…”

Section: Introductionmentioning

confidence: 99%

Generalized Fock spaces and Weyl commutation relations for the Dunkl kernel

Soltani¹

2004

Pacific J. Math.

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“…N = 1, Cholewinski [7] has investigated the SegalBargmann transform only on the Hilbert space of even functions in F k (C), by employing another approach. See also [41].…”

Section: Introductionmentioning

confidence: 99%

Segal-Bargmann transforms associated with finite Coxeter groups

Saïd

Ørsted

2005

Math. Ann.

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Using a polarization of a suitable restriction map, and heat-kernel analysis, we construct a generalized Segal-Bargmann transform associated with every finite Coxeter group G on R N . We find the integral representation of this transform, and we prove its unitarity. To define the Segal-Bargmann transform, we introduce a Hilbert space F k (C N ) of holomorphic functions on C N with reproducing kernel equal to the Dunkl-kernel. The definition and properties of F k (C N ) extend naturally those of the well-known classical Fock space. The generalized Segal-Bargmann transform allows to exhibit some relationships between the Dunkl theory in the Schrödinger model and in the Fock model. Further, we prove a branching decomposition of F k (C N ) as a unitary G × SL(2, R)-module and a general version of Hecke's formula for the Dunkl transform. (2000): 33C52, 43A85, 44A15 Mathematics Subject Classification

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“…The harmonic analysis associated with this operator and applications of this theory have been studied in many papers (see [4], [7], [9], [13], [15], [17] and [19]). …”

Section: Introductionmentioning

confidence: 99%