A positive radial product formula for the Dunkl kernel
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Abstract:
It is an open conjecture that generalized Bessel functions associated with root systems have a positive product formula for nonnegative multiplicity parameters of the associated Dunkl operators. In this paper, a partial result towards this conjecture is proven, namely a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel. Radial here means that one of the factors in the product formula is replaced by its mean over a sphere. The key to this product formula is a positivity result for the Dunkl-type spherical mean operator. It can also be interpreted in the sense that the Dunkl-type generalized translation of radial functions is positivity-preserving. As an application, we construct Dunkl-type homogeneous Markov processes associated with radial probability distributions.References
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Additional Information
- Margit Rösler
- Affiliation: Mathematisches Institut, Universität Göttingen, Bunsenstrasse 3–5, D-37073 Göttingen, Germany
- MR Author ID: 312683
- Email: roesler@uni-math.gwdg.de
- Received by editor(s): October 2, 2002
- Published electronically: January 14, 2003
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 2413-2438
- MSC (2000): Primary 33C52; Secondary 44A35, 35L15
- DOI: https://doi.org/10.1090/S0002-9947-03-03235-5
- MathSciNet review: 1973996