常曲率空间中球面平均的反演公式
Inversion Formulas for the Spherical Means in Constant Curvature Spaces
摘要：
本文的工作进一步发展了以下反演问题的理论，该问题在热声层析成像的快速发展领域中起着核心作用，并与偏微分方程和积分几何有着密切的联系：{ \it重建一个在$n$维球$B$中支持的函数$f$，如果$f$的球面平均值在所有以$B$边界为中心的测地球上都是已知的。}基于解析延拓的思想，我们提出了一种新的统一方法。这种方法不仅给出了欧氏空间$\bbr^n$的显式反演公式（如原始设置），而且给出了任意常曲率空间$X$的显式反演公式，包括$n$维球面和双曲空间。将所得结果应用于任意维常曲率空间中一类Euler-Poisson-Darboux方程的反问题。
英文摘要：
The work develops further the theory of the following inversion problem, which plays the central role in the rapidly developing area of thermoacoustic tomography and has intimate connections with PDEs and integral geometry: {\it Reconstruct a function $f$ supported in an $n$-dimensional ball $B$, if the spherical means of $f$ are known over all geodesic spheres centered on the boundary of $B$.} We propose a new unified approach based on the idea of analytic continuation. This approach gives explicit inversion formulas not only for the Euclidean space $\bbr^n$ (as in the original set-up) but also for arbitrary constant curvature space $X$, including the $n$-dimensional sphere and the hyperbolic space. The results are applied to inverse problems for a large class of Euler-Poisson-Darboux equations in constant curvature spaces of arbitrary dimension.
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