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 OLE DB Technical Materials. OLE DB White Papers
, 2001
"... Abstract. We conjecture a geometrical form of the Paley–Wiener theorem for the Dunkl transform and prove three instances thereof, by using a reduction to the onedimensional even case, shift operators, and a limit transition from Opdam’s results for the graded Hecke algebra, respectively. These Pale ..."
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Abstract. We conjecture a geometrical form of the Paley–Wiener theorem for the Dunkl transform and prove three instances thereof, by using a reduction to the onedimensional even case, shift operators, and a limit transition from Opdam’s results for the graded Hecke algebra, respectively. These Paley– Wiener theorems are used to extend Dunkl’s intertwining operator to arbitrary smooth functions. Furthermore, the connection between Dunkl operators and the Cartan motion group is established. It is shown how the algebra of radial parts of invariant differential operators can be described explicitly in terms of Dunkl operators. This description implies that the generalized Bessel functions coincide with the spherical functions. In this context of the Cartan motion group, the restriction of Dunkl’s intertwining operator to the invariants can be interpreted in terms of the Abel transform. We also show that, for certain values of the multiplicities of the restricted roots, the Abel transform is essentially inverted by a differential operator. 1. Introduction and
On SegalBargmann analysis for finite Coxeter groups and its heat kernel
, 903
"... We prove identities involving the integral kernels of three versions (two being introduced here) of the SegalBargmann transform associated to a finite Coxeter group acting on a finite dimensional, real Euclidean space (the first version essentially having been introduced around the same time by Ben ..."
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We prove identities involving the integral kernels of three versions (two being introduced here) of the SegalBargmann transform associated to a finite Coxeter group acting on a finite dimensional, real Euclidean space (the first version essentially having been introduced around the same time by Ben Saïd and Ørsted and independently by Soltani) and the Dunkl heat kernel, due to Rösler, of the Dunkl Laplacian associated with the same Coxeter group. All but one of our relations are originally due to Hall in the context of standard SegalBargmann analysis on Euclidean space. Hall’s results (trivial Dunkl structure and arbitrary finite dimension) as well as our own results in µdeformed quantum mechanics (nontrivial Dunkl structure, dimension one) are particular cases of the results proved here. So we can understand all of these versions of the SegalBargmann transform associated to a Coxeter group as Hall type transforms. In particular, we define an analogue of Hall’s Version C generalized SegalBargmann transform which is then shown to be Dunkl convolution with the Dunkl heat kernel followed by analytic continuation. In the context of Version C we also introduce a new SegalBargmann space and a new transform associated to the Dunkl theory. Also we have what appears to be a new relation in this context between the SegalBargmann kernels for
The µdeformed SegalBargmann transform as a Hall type transform
"... We present an explanation of how the µdeformed SegalBargmann spaces, that are studied in various articles of the author in collaboration with Angulo, Echevarría and Pita, can be viewed as deserving their name, that is, how they should be considered as a part of SegalBargmann analysis. This explan ..."
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We present an explanation of how the µdeformed SegalBargmann spaces, that are studied in various articles of the author in collaboration with Angulo, Echevarría and Pita, can be viewed as deserving their name, that is, how they should be considered as a part of SegalBargmann analysis. This explanation relates the µdeformed SegalBargmann transforms to the generalized SegalBargmann transforms introduced by B. Hall using heat kernel analysis. All the versions of the µdeformed SegalBargmann transform can be understood as Hall type transforms. In particular, we define a µdeformation of Hall’s “Version C ” generalized SegalBargmann transform which is then shown to be a µdeformed convolution with a µdeformed heat kernel followed by analytic continuation. Our results are generalizations and analogues of the results of Hall. Keywords: SegalBargmann analysis, heat kernel analysis, µdeformed quantum mechanics.
First Hitting Time of the Boundary of the Weyl Chamber by Radial Dunkl Processes ⋆
"... Abstract. We provide two equivalent approaches for computing the tail distribution of the first hitting time of the boundary of the Weyl chamber by a radial Dunkl process. The first approach is based on a spectral problem with initial value. The second one expresses the tail distribution by means of ..."
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Abstract. We provide two equivalent approaches for computing the tail distribution of the first hitting time of the boundary of the Weyl chamber by a radial Dunkl process. The first approach is based on a spectral problem with initial value. The second one expresses the tail distribution by means of the Winvariant Dunkl–Hermite polynomials. Illustrative examples are given by the irreducible root systems of types A, B, D. The paper ends with an interest in the case of Brownian motions for which our formulae take determinantal forms. Key words: radial Dunkl processes; Weyl chambers; hitting time; multivariate special functions; generalized Hermite polynomials 2000 Mathematics Subject Classification: 33C20; 33C52; 60J60; 60J65 1
Imaginary Powers of the Dunkl Harmonic Oscillator ⋆
, 2008
"... doi:10.3842/SIGMA.2009.016 Abstract. In this paper we continue the study of spectral properties of the Dunkl harmonic oscillator in the context of a finite reflection group on R d isomorphic to Z d 2. We prove that imaginary powers of this operator are bounded on L p, 1 < p < ∞, and from L 1 into we ..."
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doi:10.3842/SIGMA.2009.016 Abstract. In this paper we continue the study of spectral properties of the Dunkl harmonic oscillator in the context of a finite reflection group on R d isomorphic to Z d 2. We prove that imaginary powers of this operator are bounded on L p, 1 < p < ∞, and from L 1 into weak L 1.
Symmetry, Integrability and Geometry: Methods and Applications External Ellipsoidal Harmonics for the Dunkl–Laplacian ⋆
"... Abstract. The paper introduces external ellipsoidal and external spheroconal hharmonics for the Dunkl–Laplacian. These external hharmonics admit integral representations, and they are connected by a formula of Niven’s type. External hharmonics in the plane are expressed in terms of Jacobi polyno ..."
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Abstract. The paper introduces external ellipsoidal and external spheroconal hharmonics for the Dunkl–Laplacian. These external hharmonics admit integral representations, and they are connected by a formula of Niven’s type. External hharmonics in the plane are expressed in terms of Jacobi polynomials P α,β n kind. and Jacobi’s functions Q α,β n of the second Key words: external ellipsoidal harmonics; Stieltjes polynomials; Dunkl–Laplacian; fundamental solution; Niven’s formula; Jacobi’s function of the second kind 2000 Mathematics Subject Classification: 33C52; 35C10 1
Liouville Theorem for Dunkl Polyharmonic Functions ⋆
, 2008
"... Original article is available at ..."
Symmetry, Integrability and Geometry: Methods and Applications Generalized Bessel function of Type D ⋆
"... Abstract. We write down the generalized Bessel function associated with the root system of type D by means of multivariate hypergeometric series. Our hint comes from the particular case of the Brownian motion in the Weyl chamber of type D. Key words: radial Dunkl processes; Brownian motions in Weyl ..."
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Abstract. We write down the generalized Bessel function associated with the root system of type D by means of multivariate hypergeometric series. Our hint comes from the particular case of the Brownian motion in the Weyl chamber of type D. Key words: radial Dunkl processes; Brownian motions in Weyl chambers; generalized Bessel function; multivariate hypergeometric series 2000 Mathematics Subject Classification: 33C20; 33C52; 60J60; 60J65 1 Root systems and related processes We refer the reader to [11] for facts on root systems. Let (V, 〈·〉) be an Euclidean space of finite dimension m ≥ 1. A reduced root system R is a finite set of non zero vectors in V such that 1) R ∩ Rα = {α, −α} for all α ∈ R, 2) σα(R) = R, where σα is the reflection with respect to the hyperplane Hα orthogonal to α 〈α, x〉 σα(x) = x − 2 α, x ∈ V. 〈α, α〉
Symmetry, Integrability and Geometry: Methods and Applications From slq(2) to a Parabosonic Hopf Algebra
, 2011
"... Abstract. A Hopf algebra with four generators among which an involution (reflection) operator, is introduced. The defining relations involve commutators and anticommutators. The discrete series representations are developed. Designated by sl−1(2), this algebra encompasses the Lie superalgebra osp(1 ..."
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Abstract. A Hopf algebra with four generators among which an involution (reflection) operator, is introduced. The defining relations involve commutators and anticommutators. The discrete series representations are developed. Designated by sl−1(2), this algebra encompasses the Lie superalgebra osp(12). It is obtained as a q = −1 limit of the slq(2) algebra and seen to be equivalent to the parabosonic oscillator algebra in irreducible representations. It possesses a noncocommutative coproduct. The Clebsch–Gordan coefficients (CGC) of sl−1(2) are obtained and expressed in terms of the dual −1 Hahn polynomials. A generating function for the CGC is derived using a Bargmann realization. Key words: parabosonic algebra; dual Hahn polynomials; Clebsch–Gordan coefficients 2010 Mathematics Subject Classification: 17B37; 17B80; 33C45
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, 707
"... Direct and reverse logSobolev inequalities in µdeformed SegalBargmann analysis ..."
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Direct and reverse logSobolev inequalities in µdeformed SegalBargmann analysis