Dunkl Kernel Research Articles

Sonine-type integral transforms related to the Dunkl operator on the real line

We consider on ℝ the differential-difference operator which is referred to as the Dunkl operator with parameter γ+1/2 associated with the reflection ℤ2 on ℝ. We exhibit a Sonine-type integral representation for the Dunkl kernel e γ, eigenfunction of the Dunkl operator Λγ. This enables us to construct a new Sonine integral transform on ℝ tied to Λγ, which turns out together with its dual to be transmutation operators between two Dunkl operators of different parameters.

Sonine Transform Associated to the Dunkl Kernel on the Real Line

We consider the Dunkl intertwining operator V and its dual t V , we define and study the Dunkl Sonine operator and its dual on R. Next, we introduce complex powers of the Dunkl Laplacian and establish inversion formulas for the Dunkl Sonine operatorS, and its dual t S, . Also, we give a Plancherel formula for the operator t S, .

The Dunkl intertwining operator

We consider Dunkl theory associated to a general Coxeter group G. A new characterization of the regularity of the weight k is given and a new construction, devoid of Kozul complex theory, of the Dunkl intertwining operator V k is established. We apply our results to derive sharp estimates of the Dunkl kernel. We give explicit formula in the case of orthogonal positive root systems.

Generalized Fock spaces and Weyl commutation relations for the Dunkl kernel

In this paper we study a class of generalized Fock spaces associated with the Dunkl operators. Next we introduce the commutator relations between the Dunkl operators and multiplication operators which lead to a generalized class of Weyl commutation relations for the Dunkl kernel.

Positivity of Dunkl's Intertwining Operator via the Trigonometric Setting

AbstractIn this note, a new proof for the positivity of Dunkl’s intertwining operator in thecrystallographic case is given. It is based on an asymptotic relationship between theOpdam-Cherednik kernel and the Dunkl kernel as recently observed by M. de Jeu, andon positivity results of S. Sahi for the Heckman-Opdam polynomials and their non-symmetric counterparts. 2000 AMS Subject Classification: 33C52, 33C67. 1 Introduction In [R], it was proven that Dunkl’s intertwining operator between the rational Dunkl operatorsfor a fixed finite reflection group and nonnegative multiplicity function is positive. As aconsequence we obtained an abstract Harish-Chandra type integral representation for theDunkl kernel, the image of the usual exponential kernel under the intertwiner. The proofwas based on methods from the theory of operator semigroups and a rank-one reduction.In the present note, we give a new, completely different proof of these results under theonly additional assumption that the underlying reflection group has to be crystallographic.In contrast to the proof of [R], where precise information on the supports of the representingmeasures could only be obtained by going back to estimates of the kernel from [J1], thisinformation is now directly obtained. Our new approach relies first on an asymptotic rela-tionship between the Opdam-Cherednik kernel and the Dunkl kernel as recently observed byde Jeu [J2], and second on positivity results of Sahi [S] for the Heckman-Opdam polynomialsand their non-symmetric counterparts.

POSITIVITY OF THE INTERTWINING OPERATOR AND HARMONIC ANALYSIS ASSOCIATED WITH THE JACOBI-DUNKL OPERATOR ON ${\mathbb R}$

We consider a differential-difference operator Λα,β, [Formula: see text], [Formula: see text] on [Formula: see text]. The eigenfunction of this operator equal to 1 at zero is called the Jacobi–Dunkl kernel. We give a Laplace integral representation for this function and we prove that for [Formula: see text], [Formula: see text], the kernel of this integral representation is positive. This result permits us to prove that the Jacobi–Dunkl intertwining operator and its dual are positive. Next we study the harmonic analysis associated with the operator Λα,β (Jacobi–Dunkl transform, Jacobi–Dunkl translation operators, Jacobi–Dunkl convolution product, Paley–Wiener and Plancherel theorems…).

A positive radial product formula for the Dunkl kernel

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.

Asymptotic Analysis for the Dunkl Kernel

This paper studies the asymptotic behavior of the integral kernel of the Dunkl transform, the so-called Dunkl kernel, when one of its arguments is fixed and the other tends to infinity either within a Weyl chamber of the associated reflection group, or within a suitable complex domain. The obtained results are based on the asymptotic analysis of an associated system of ordinary differential equations. They generalize the well-known asymptotics of the confluent hypergeometric function 1F1 to the higher-dimensional setting and include a complete short-time asymptotics for the Dunkl-type heat kernel. As an application, it is shown that the representing measures of Dunkl's intertwining operator are generically continuous.

Generalized Fock spaces and Weyl relations for the Dunkl kernel on the real line

In this paper we study a class of generalized Fock spaces associated with the Dunkl operator. Next we introduce the commutator relations between the Dunkl operator and multiplication operator which leads to a generalized class of Weyl relations for the Dunkl kernel.