Segal-Bargmann transforms associated with finite Coxeter groups

Saïd, Salem Ben; Ørsted, Bent

doi:10.1007/s00208-005-0718-3

Cited by 27 publications

(61 citation statements)

References 35 publications

(60 reference statements)

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“…Now we can show that the integral of x, t + is k y, t − is k over St (1) is indeed the reproducing kernel K k (x, y). Note that this result was already established in [4] in a different way in the framework of an inverse Szegő-Radon projection.…”

Section: Plane Wave Formulasmentioning

confidence: 91%

“…A Pizzetti formula for Stiefel manifolds was developed in [5]. We state here the result for the first case St (1) .…”

Section: Plane Wave Formulasmentioning

confidence: 96%

“…Mapping a given k-homogeneous polynomial to its harmonic component of degree k − 2ℓ can be done by means of the projection operator (see e.g. [1] and [15])…”

Section: Spherical Harmonicsmentioning

confidence: 99%

“…The tuple (t, s) thus lies on a Stiefel manifold of order 2. We consider two types of Stiefel manifolds that we denote as St (1) and St (2) . These are isomorphic to the homogeneous spaces…”

Section: Plane Wave Formulasmentioning

confidence: 99%

“…Note that in the above expression St (1) dσ(s)dσ(t) can be interpreted as the integral over the (m − 1)dimensional sphere S m−1 with respect to t and the integral over the (m − 2)-dimensional subsphere S m−2 ⊥ with respect to s, that is perpendicular to t ∈ S m−1 . Our aim is to integrate the plane wave x, z k y,z k over the manifold St (1) to show that the result equals the reproducing kernel K k (x, y) of the spherical harmonics. In order to apply the previous theorem it is necessary to know the actions of the operators I 1 and I 2 on these plane waves.…”

Section: Plane Wave Formulasmentioning

confidence: 99%

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Plane wave formulas for spherical, complex and symplectic harmonics

Bie

Sommen

Wutzig

2017

Journal of Approximation Theory

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This paper is concerned with spherical harmonics, and two refinements thereof: complex harmonics and symplectic harmonics. The reproducing kernels of the spherical and complex harmonics are explicitly given in terms of Gegenbauer or Jacobi polynomials. In the first part of the paper we determine the reproducing kernel for the space of symplectic harmonics, which is again expressible as a Jacobi polynomial of a suitable argument. In the second part we find plane wave formulas for the reproducing kernels of the three types of harmonics, expressing them as suitable integrals over Stiefel manifolds. This is achieved using Pizzetti formulas that express the integrals in terms of differential operators

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Section: Plane Wave Formulasmentioning

confidence: 91%

“…A Pizzetti formula for Stiefel manifolds was developed in [5]. We state here the result for the first case St (1) .…”

Section: Plane Wave Formulasmentioning

confidence: 96%

“…Mapping a given k-homogeneous polynomial to its harmonic component of degree k − 2ℓ can be done by means of the projection operator (see e.g. [1] and [15])…”

Section: Spherical Harmonicsmentioning

confidence: 99%

Section: Plane Wave Formulasmentioning

confidence: 99%

Section: Plane Wave Formulasmentioning

confidence: 99%

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Plane wave formulas for spherical, complex and symplectic harmonics

Bie

Sommen

Wutzig

2017

Journal of Approximation Theory

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An Extension of Pizzetti’s Formula Associated with the Dunkl Operators

Shimeno

Tani

2021

Springer Proceedings in Mathematics &Amp; Statistics

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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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Segal-Bargmann transforms associated with finite Coxeter groups

Cited by 27 publications

References 35 publications

Plane wave formulas for spherical, complex and symplectic harmonics

Plane wave formulas for spherical, complex and symplectic harmonics

An Extension of Pizzetti’s Formula Associated with the Dunkl Operators

Relations Among Various Versions of the Segal-Bargmann Transform

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