Generalized Fock spaces and Weyl commutation relations for the Dunkl kernel

Soltani, Fethi

doi:10.2140/pjm.2004.214.379

Cited by 22 publications

(25 citation statements)

References 18 publications

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“…The statement of Theorem 1 in [42] matches our Theorem 4.2, whilst our Theorem 3.2 is slightly different from Proposition 1 of [42], since we prove that the functions in the generalized Fock space are holomorphic, while in [42] the space was defined as a completion of a subset of holomorphic functions. Proposition 4, Lemma 5, and Proposition 5 of [42] together form our Theorem 3.7. Notice though that our construction and approach are different from the ones used by Soltani, who employs a similar argument to that used earlier by Cholewinski [7] in the rank one case.…”

Section: Introductionsupporting

confidence: 62%

“…After this manuscript was completed, we learned about the paper [42] by Soltani, where some results overlap with ours. (During the time of writing this paper, [42] was in the preprint stage.)…”

Section: Introductionmentioning

confidence: 73%

“…(During the time of writing this paper, [42] was in the preprint stage.) The statement of Theorem 1 in [42] matches our Theorem 4.2, whilst our Theorem 3.2 is slightly different from Proposition 1 of [42], since we prove that the functions in the generalized Fock space are holomorphic, while in [42] the space was defined as a completion of a subset of holomorphic functions. Proposition 4, Lemma 5, and Proposition 5 of [42] together form our Theorem 3.7.…”

Section: Introductionmentioning

confidence: 99%

“…Notice though that our construction and approach are different from the ones used by Soltani, who employs a similar argument to that used earlier by Cholewinski [7] in the rank one case. We thank M. Rösler and M. Sifi for bringing the reference [42] to our attention. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionsupporting

confidence: 62%

Section: Introductionmentioning

confidence: 73%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Segal-Bargmann transforms associated with finite Coxeter groups

Saïd

Ørsted

2005

Math. Ann.

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“…They are important in pure mathematics and in certain parts of quantum mechanics and one expects that the results in this paper will be useful when discussing boundedness properties in Dunkl analysis. Furthermore, these operators provide a useful tool in the study of special functions associated with root systems [8,9,19,23]. They are closely related to certain representations of degenerated affine Hecke algebras [5,13].…”

Section: Introductionmentioning

confidence: 99%

Littlewood–Paley operators associated with the Dunkl operator on R

Soltani¹

2005

Journal of Functional Analysis

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Relations Among Various Versions of the Segal-Bargmann Transform

Sontz

2012

Recent Progress in Operator Theory and Its Applications

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We present various relations among Versions A, B and C of the Segal-Bargmann transform. We get results for the Segal-Bargmann transform associated to a Coxeter group acting on a finite dimensional Euclidean space. Then analogous results are shown for the Segal-Bargmann transform of a connected, compact Lie group for all except one of the identities established in the Coxeter case. A counterexample is given to show that the remaining identity from the Coxeter case does not have an analogous identity for the Lie group case. A major result is that in both contexts the Segal-Bargmann transform for Version C is determined by that for Version A. (2000): Primary 45H05, 44A15; Secondary 46E15 Keywords: Segal-Bargmann transfrom, Coxeter group, Dunkl heat kernel 1 Research partially supported by CONACYT (Mexico) project 49187. Mathematics Subject ClassificationWe will use the holomorphic Dunkl kernel function E µ :We will also be using the analytic continuation of the Dunkl heat kernel, which is given for z, w ∈ C N by( 1.1) This kernel arises in the solution of the initial value problem of the heat equation associated with the Dunkl Laplacian operator. (See [13].) We next define the kernel functions of the versions of the Segal-Bargmann transform associated to a Coxeter group for z ∈ C N and q ∈ R N by). (1.4) See [1], [4] and [15] for the origins of this theory in the case µ ≡ 0. The versions of the Segal-Bargmann transform are given as follows. (See [2], [5], [16], [17] and [20].) Versions A and C are defined byand ω µ,t is the density of a measure on R N . Version B is defined bywhere z ∈ C N , f ∈ L 2 (R N , m µ,t ) and m µ,t is the density of a measure on R N . Associated to these versions there are reproducing kernel Hilbert spaces of holomorphic functions f : C N → C, denoted A µ,t , B µ,t and C µ,t respectively, such that A µ,t : L 2 (R N , ω µ,t ) → A µ,t B µ,t : L 2 (R N , m µ,t ) → B µ,t C µ,t : L 2 (R N , ω µ,t ) → C µ,t are unitary isomorphisms. It turns out that A µ,t = B µ,t as Hilbert spaces.

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Generalized Fock spaces and Weyl commutation relations for the Dunkl kernel

Abstract: In this paper we study a class of generalized Fock spaces associated with the Dunkl operators. Next we introduce the commutator relations between the Dunkl operators and multiplication operators which lead to a generalized class of Weyl commutation relations for the Dunkl kernel.

Cited by 22 publications

References 18 publications

Segal-Bargmann transforms associated with finite Coxeter groups

Segal-Bargmann transforms associated with finite Coxeter groups

Littlewood–Paley operators associated with the Dunkl operator on R

Relations Among Various Versions of the Segal-Bargmann Transform

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