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Positivity of the Jacobi-Cherednik intertwining operator and its dual. (English) Zbl 1208.47027

The authors study the Jacobi-Cherednik operator \(T^{(k,k^{\prime})}f(x)\), \(k>0\), \(k^{\prime}\geq 0\). The operator \(T^{(k,k^{\prime})}f(x)\) is a special case, precisely the one-dimensional case, of the Cherednik operator \(T_{\xi}f(x)\). They show that there exists a continuous kernel (Laplace kernel) for its eigenvalues, which satisfies some positivity properties. They also study the associated intertwining operator \(V^{(k,k^{\prime})}f(x)\) and the intertwining dual operator \(^{t}V^{(k,k^{\prime})}f(x)\).

MSC:

47B34 Kernel operators
44A15 Special integral transforms (Legendre, Hilbert, etc.)
47B39 Linear difference operators
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