A positive radial product formula for the Dunkl kernel

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.

Bessel convolutions on matrix cones

“…In the rank-one case there is such a convolution, but it is not positivity-preserving. For details and affirmative results in this direction see [Rös03a]. It is, however, conjectured that for arbitrary G and k 0, the Bessel functions J k have a positive product formula which leads to a commutative hypergroup structure on a distinguished closed Weyl chamber Ξ of G, the dual of this hypergroup being made up by the functions ξ → J k (ξ, η), η ∈ Ξ.…”

“…It is, however, conjectured that for arbitrary G and k 0, the Bessel functions J k have a positive product formula which leads to a commutative hypergroup structure on a distinguished closed Weyl chamber Ξ of G, the dual of this hypergroup being made up by the functions ξ → J k (ξ, η), η ∈ Ξ. In rank one and in all Cartan motion group cases this is true, see [Rös03a]. In the following, we confirm this conjecture for three continuous series of multiplicities for root system B q .…”

“…The value of d µ can be calculated explicitly; it is a particular case of a Selberg type integral which was evaluated by Macdonald [Mac82] for the classical root systems. In the general Dunkl setting, there is a generalized translation on suitable function spaces which replaces the usual group addition to some extent, see [Rös98,Rös03a] and the references cited there. On L 2 (R q , w k ), this translation is defined by…”

“…The three continuous series (d = 1, 2, 4) for B q obtained in this paper are, to the best of our knowledge, the first affirmative examples beyond the group cases associated with Cartan motion groups of reductive symmetric spaces. Some background, together with further partial results, is given in [Rös03a].…”

The Clifford Deformation of the Hermite Semigroup

“…Remark 7.4. For the Dunkl transform (see, e.g., [24,28]) and for the Clifford-Fourier transform (see [8]) one can compute even a more general integral of the form R m K y, x; i π 2 K z, y; −i π 2 f (r y )h(r y )dy with f (r y ) an arbitrary radial function of suitable decay. This is done by using the addition formula for the Bessel function…”