Korean J. Math. Vol. 27 No. 1 (2019) pp.221-267
DOI: https://doi.org/10.11568/kjm.2019.27.1.221

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

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Published: Mar 30, 2019
Keywords:
Cherednik's operators on $\mathbb{R}^d$, Heckman-Opdam's theory on $\mathbb{R}^d$, hypergeometric transmutation operators, Hypergeometric translation operator, Hypergeometric convolution product, Heckman-Opdam's hypergeometric function, Hypergeometri
Mathematics Subject Classification:
33E30, 51F15, 33C67, 43A32, 43A62.

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Khalifa Trimeche
Faculty of Sciences of Tunis, Department of Mathematics, CAMPUS, 2092 Tunis, Tunisia

Abstract

In the five first sections of this paper we define and study the hypergeometric transmutation operators $V^W_k$ and ${}^tV^W_k$ 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 $\mathcal{T}^W_x, x \in \mathbb{R}^d$, and its dual ${}^t\mathcal{T}^W_x, x \in \mathbb{R}^d$, 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 $L^p_{\mathcal{A}_k}$-spaces, and we obtain some interesting results. In the sixth section we consider a some root system of type $BC_d$ (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.


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