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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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.
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.
참고문헌 (23)
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