Positivity of Dunkl’s intertwining operator

Rösler, Margit

doi:10.1215/s0012-7094-99-09813-7

Cited by 242 publications

(174 citation statements)

References 19 publications

(7 reference statements)

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“…A concrete description of this intertwiner operator V k is presently still unknown, with the exception of the onedimensional case [8] and the case A 2 [11]. Significant abstract results were obtained by Rösler [30], who showed, amongst others, that V k is for k ≥ 0 a positive operator which can be described in terms of measures.…”

Section: Notations and Previous Resultsmentioning

confidence: 99%

Paley–Wiener theorems for the Dunkl transform

Jeu

2006

Trans. Amer. Math. Soc.

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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 one-dimensional even case, shift operators, and a limit transition from Opdam's results for the graded Hecke algebra, respectively. These PaleyWiener 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.

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Section: Notations and Previous Resultsmentioning

confidence: 99%

Paley–Wiener theorems for the Dunkl transform

Jeu

2006

Trans. Amer. Math. Soc.

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“…This kernel has a unique analytic extension to C d × C d (see [7]). The Dunkl kernel has the Laplace-type representation [8] …”

Section: The Dunkl-wigner Transformmentioning

confidence: 99%

Reproducing inversion formulas for the Dunkl-Wigner transforms

Soltani

2015

Cubo

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We define and study the Fourier-Wigner transform associated with the Dunkl operators, and we prove for this transform a reproducing inversion formulas and a Plancherel formula. Next, we introduce and study the extremal functions associated to the DunklWigner transform. RESUMENDefinimos y estudiamos la transformada de Fourier-Wigner asociada a los operadores de Dunkl, y probamos una fórmula de inversion y una formula de Plancherel para esta transformada. Luego introducimos y estudiamos las funciones extramales asociadas a la transformada de Dunkl-Wigner.

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“…). Moreover it is a positive operator (see [18]) and can be extended to the space of smooth functions and even to the space of distributions (see [29,30]). …”

Section: Some Facts In Dunkl's Theorymentioning

confidence: 99%