A positive radial product formula for the Dunkl kernel

Rösler, Margit

doi:10.1090/S0002-9947-03-03235-5

A positive radial product formula for the Dunkl kernel
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Trans. Amer. Math. Soc. 355 (2003), 2413-2438 Request permission

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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