In the hyperbolic disc (or more generally in real hyperbolic spaces) we consider the horospherical Radon transform R and the geodesic Radon transform X. Composition with their respective dual operators yields two convolution operators on the disc (with respect to the hyperbolic measure). We describe their convolution kernels in comparison with those of the corresponding operators on a homogeneous tree T, separately studied as acting on functions on the vertices or on the edges. This leads to a new theory of spherical functions and Radon inversion on the edges of a tree.

中文翻译：

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双曲空间中的Radon变换及其离散对应

在双曲盘（或更一般在实际双曲空间）我们认为horospherical Radon变换[R和测地Radon变换X。用它们各自的对偶运算符进行合成会在光盘上产生两个卷积运算符（关于双曲度量）。我们将它们的卷积核与同构树T上的相应算符进行比较，对它们的卷积核进行单独研究，以研究它们在顶点或边缘上的作用。这导致了球函数和树边缘Radon求逆的新理论。