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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.
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참고문헌 (23)
참고문헌 신청
W.R.Bloom / 1995 / Harmonic Analysis of Probability Measures on Hy-pergroups / Walter de Gruyter
I. Cherednik / 1991 / Aunifications of Knizhnik Zamolodchnikov equations and Dunkl operators via affine Hecke algebras / Invent. Math 106:411~432
I.Cherednik / 1997 / Inverse Harish-Chandra transform and difference operators / Int. Math. Res. Not 15:733~750
Deepmala / 2014 / A Study on Fixed Point Theorems for Nonlinear Contractions and its Applications
C.F.Dunkl / 1973 / The measure algebra of a locally compact hypergroup / Trans. Amer. Math. Soc 179:331~348
I. Cherednik / 1991 / Aunifications of Knizhnik Zamolodchnikov equations and Dunkl operators via affine Hecke algebras / Invent. Math 106:411~432
I.Cherednik / 1997 / Inverse Harish-Chandra transform and difference operators / Int. Math. Res. Not 15:733~750
Deepmala / 2014 / A Study on Fixed Point Theorems for Nonlinear Contractions and its Applications
C.F.Dunkl / 1973 / The measure algebra of a locally compact hypergroup / Trans. Amer. Math. Soc 179:331~348
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