A positive radial product formula for the Dunkl kernel

By M. Rösler. 2002

243 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4ae0d7698a1817e52ccec9273f539534f2fc5181

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Citations in the corpus

The Class of Clifford-Fourier Transforms (2011, unknown): “[10, 27]).”
Convolution operator and maximal function for the Dunkl transform (2004, background): “The basic properties of the Dunkl transform have been studied in [3, 8, 13, 15] and also in [12, 19] (see also the references therein).”
An Introduction to Dunkl Theory and Its Analytic Aspects (2016, background): “In general there is a lack information about (35) and the following facts are conjectured [68].”
A transference theorem for the Dunkl transform and its applications (2010, methodology): “8) was first proved in [14] under the assumption that f is a radial Schwartz function.”
Continuous crystal and Duistermaat-Heckman measure for Coxeter groups (2008, background): “The positive product formula gives a positive answer to a question of Rösler [35] for the radial Dunkl kernel.”
Positive $$L^p$$Lp-Bounded Dunkl-Type Generalized Translation Operator and Its Applications (2017, unknown): “(3.1), we get Z Rd τ−yf(x)g(y)dµ k(y) = Z Rd g(y) Z Rd ek(−y,z)ek(x,z)Fk(f)(z)dµk(z)dµk(y) = Z Rd ek(x,z)Fk(f)(z)Fk(g)(z)dµk(z). and the second part in (3.14) is valid. Let y∈ Rd be given. M. Ro¨sler [43] proved that the operator τy is positive on C∞ rad(R d), i.e., τy ≥ 0, and moreover, for any x∈ ”