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Trimeche, Khalifa The Kangwon-Kyungki Mathematical Society 2014 한국수학논문집 Vol.22 No.1
We consider the hypergeometric translation operator associated to the Cherednik operators and the Heckman-Opdam theory attached to the root system of type $B_2$. We prove in this paper that these operators are positivity preserving and allow positive integral representations. In particular we deduce that the product formulas of the Opdam-Cherednik and the Heckman-Opdam kernels are positive integral transforms, and we obtain best estimates of these kernels. The method used to obtain the previous results shows that these results are also true in the case of the root system of type $C_2$.
Khalifa Trimeche 강원경기수학회 2019 한국수학논문집 Vol.27 No.1
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.
Trimeche, Khalifa The Kangwon-Kyungki Mathematical Society 2014 한국수학논문집 Vol.22 No.4
We prove in this paper the absolute continuity of the representing measures of the hypergeometric translation operators $\mathcal{T}_x$ and $\mathcal{T}_x^W$ associated respectively to the Cherednik operators and the Heckman-Opdam theory attached to the root system of type $B_2$ and $C_2$ which are studied in [9].
Amina Hassini,Khalifa Trimeche 강원경기수학회 2020 한국수학논문집 Vol.28 No.4
In this paper we give the harmonic analysis associated with the Cherednik operators, next we define and study the Cherednik wavelets and the Cherednik windowed transforms on $\mathbb{R}^d$, in the W-invariant case, and we prove for these transforms Plancherel and inversion formulas. As application we give these results for the Gaussian Cherednik wavelets and the Gaussian Cherednik windowed transform on $\mathbb{R}^d$ in the W-invariant case.
Hassini, Amina,Maalaoui, Rayaane,Trimeche, Khalifa The Kangwon-Kyungki Mathematical Society 2016 한국수학논문집 Vol.24 No.2
By using the Heckman-Opdam theory on ${\mathbb{R}}^d$ given in [20], we define and study in this paper, the generalized wavelets on ${\mathbb{R}}^d$ and the generalized wavelet transform on ${\mathbb{R}}^d$, and we establish their properties. Next, we prove for the generalized wavelet transform Plancherel and inversion formulas.