The harmonic analysis associated to the Heckman-Opdam's theory and its application to a root system of type BCd

Abstract

In the five first sections of this paper we define and study the hypergeometric transmutation operators $V_k^W$ and ${}^t{V}_k^W$ called also the trigonometric Dunkl intertwining operator and its dual corresponding to the Heckman-Opdam's theory on ${\mathbb{R}}^d$. By using these operators we define the hypergeometric translation operator ${\mathcal{T}}_x^W,x\in {\mathbb{R}}^d$, and its dual ${}^t{\mathcal{T}}_x^W,x\in {\mathbb{R}}^d$, we express them in terms of the hypergeometric Fourier transform ${\mathcal{H}}^W$, we give their properties and we deduce simple proofs of the Plancherel formula and the Plancherel theorem for the transform ${\mathcal{H}}^W$. We study also the hypergeometric convolution product on $W$-invariant $L_{{\mathcal{A}}_k}^p$-spaces, and we obtain some interesting results. In the sixth section we consider a some root system of type $B{C}_d$ (see [17]) of whom the corresponding hypergeometric translation operator is a positive integral operator. By using this positivity we improve the results of the previous sections and we prove others more general results.