arXiv 2002.090992020-02-26math.FA

齐次树上的Radon变换

Radon transforms on homogeneous trees

Enrico Casadio Tarabusi, Simon G. Gindikin, Massimo A. Picardello

摘要：

我们研究了不同的horosphere Radon变换，所产生的关于一个齐次树T作为一个单纯复形，其单纯形的顶点V，边缘E或标志F（标志是定向的边缘）。端点（从参考顶点开始的无限测地线）为树提供边界$Ω$。然后，半球与基底$Ω$和纤维$\mathZ$形成平凡主纤维丛。有三个这样的纤维丛，由顶点、边或旗的半球组成，但它们是同构的：然而，这些纤维丛之间没有同构将特殊截面映射到特殊截面（特殊截面由通过给定顶点、边或旗的半球集合组成）。纤维丛的自同构群包含平行移位的子群$A$，类似于半单群的嘉当子群。T上Laplace算子的归一化本征函数是泊松核的复幂的边界积分，即$A$的特征，并且是由$A$在Mackey意义下导出的表示的矩阵系数，即所谓的球面表示.顶点-水平球Radon变换由每个顶点-水平球中的V上的求和组成，并且对于边缘或标志也类似。我们证明了所有这些Radon变换的反演公式，并给出了它们在T上的调和分析和Plancherel测度中的应用.我们通过积分几何证明了顶点和边的球面表示是等价的。此外，我们定义了Radon反投影，并通过将其与其反投影组合来找到每个Radon变换的反演算子。这产生了T上的卷积算子，其符号通过球面傅里叶变换获得，其倒数是Radon反演公式的符号。

英文摘要：

We study the different horospherical Radon transforms that arise by regarding a homogeneous tree T as a simplicial complex whose simplices are vertices V, edges E or flags F (flags are oriented edges). The ends (infinite geodesic rays starting at a reference vertex) provide a boundary $Ω$ for the tree. Then the horospheres form a trivial principal fiber bundle with base $Ω$ and fiber $\mathZ$. There are three such fiber bundles, consisting of horospheres of vertices, edges or flags, but they are isomorphic: however, no isomorphism between these fiber bundles maps special sections to special sections (a special section consists of the set of horospheres through a given vertex, edge or flag). The groups of automorphisms of the fiber bundles contain a subgroup $A$ of parallel shifts, analogous to the Cartan subgroup of a semisimple group. The normalized eigenfunctions of the Laplace operator on T are boundary integrals of complex powers of the Poisson kernel, that is characters of $A$, and are matrix coefficients of representations induced from $A$ in the sense of Mackey, the so-called spherical representations. The vertex-horospherical Radon transform consists of summation over V in each vertex-horosphere, and similarly for edges or flags. We prove inversion formulas for all these Radon transforms, and give applications to harmonic analysis and the Plancherel measure on T. We show via integral geometry that the spherical representations for vertices and edges are equivalent. Also, we define the Radon back-projections and find the inversion operator of each Radon transform by composing it with its back-projection. This gives rise to a convolution operator on T, whose symbol is obtained via the spherical Fourier transform, and its reciprocal is the symbol of the Radon inversion formula.

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