Harmonic analysis associated with the Heckman-Opdam-Jacobi operator on \(\mathbb{R}^{d+1}\). (English) Zbl 1538.42028
Summary: In this paper we consider the Heckman-Opdam-Jacobi operator \(\Delta_{HJ}\) on \(\mathbb{R}^{d+1}\). We define the Heckman-Opdam-Jacobi intertwining operator \(V_{HJ}\), which turns out to be the transmutation operator between \(\Delta_{HJ}\) and the Laplacian \(\Delta_{d+1}\). Next we construct \(^tV_{HJ}\) the dual of this intertwining operator. We exploit these operators to develop a new harmonic analysis corresponding to \(\Delta_{HJ}\).
MSC:
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 33E30 | Other functions coming from differential, difference and integral equations |
| 44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
| 35K05 | Heat equation |
Keywords:
Heckman-Opdam-Jacobi operator; generalized intertwining operator and its dual; generalized Fourier transform; generalized translation operatorsReferences:
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