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A positive radial product formula for the Dunkl kernel


Author: Margit Rösler
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
Published electronically: January 14, 2003
MathSciNet review: 1973996
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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.


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Additional Information

Margit Rösler
Affiliation: Mathematisches Institut, Universität Göttingen, Bunsenstrasse 3–5, D-37073 Göttingen, Germany
Email: roesler@uni-math.gwdg.de

DOI: https://doi.org/10.1090/S0002-9947-03-03235-5
Keywords: Dunkl operators, Dunkl kernel, product formula, multivariable Bessel functions
Received by editor(s): October 2, 2002
Published electronically: January 14, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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