Using the harmonic analysis associated to the Heckman–Opdam–Jacobi operator, relating to the root system \(BC_d\), we define and study the Wigner and Weyl transforms \(W_{\sigma }\) where \(\sigma \) is a symbol in \(S^m,m\in {\mathbb {R}}\). We give the connection between these transforms, and criterias in terms of the symbol \(\sigma \) to prove the boundedness and compactness of the transform \(W_{\sigma }\).

中文翻译：

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Wigner和Weyl变换附加到关于$$ {\ mathbb {R}} ^ {d + 1} $$ R d + 1的Heckman–Opdam–Jacobi理论

使用与根系统\（BC_d \）相关的Heckman–Opdam–Jacobi算子相关的谐波分析，我们定义并研究Wigner和Weyl变换\（W _ {\ sigma} \，其中\（\ sigma \）是\（S ^ m，m \ {\ mathbb {R}} \）中的符号。我们用符号\（\ sigma \）给出了这些变换和准则之间的联系，以证明变换\（W _ {\ sigma} \）的有界性和紧致性。