论文标题
海森堡X射线变换的注射率
Injectivity of the Heisenberg X-ray Transform
论文作者
论文摘要
我们启动了X射线断层扫描对亚及雷曼分歧的研究，为此，海森伯格组表现出最简单的非平凡示例。借助组傅立叶变换的语言，我们证明了傅立叶切片定理的操作符值的化身，并应用了此新工具，以证明Heisenberg组上足够规则的函数由其线积分在次 - 里程曼地理上的线积分确定。我们还考虑了驯服指标的家族$g_ε$近似次曼尼亚标准，并表明关联的X射线变换对所有$ε> 0 $都具有注入性。该结果给出了一个具体的示例，即具有丰富的结合点的几何形状中的注射X射线变换。
We initiate the study of X-ray tomography on sub-Riemannian manifolds, for which the Heisenberg group exhibits the simplest nontrivial example. With the language of the group Fourier Transform, we prove an operator-valued incarnation of the Fourier Slice Theorem, and apply this new tool to show that a sufficiently regular function on the Heisenberg group is determined by its line integrals over sub-Riemannian geodesics. We also consider the family of taming metrics $g_ε$ approximating the sub-Riemannian metric, and show that the associated X-ray transform is injective for all $ε>0$. This result gives a concrete example of an injective X-ray transform in a geometry with an abundance of conjugate points.
