For a one-parameter family of simple metrics of constant curvature ($4\kappa$ for $\kappa\in (-1,1)$) on the unit disk $M$, we first make explicit the Pestov–Uhlmann range characterization of the geodesic X-ray transform, by constructing a basis of functions making up its range and co-kernel. Such a range characterization also translates into moment conditions $à$ $la$ Helgason–Ludwig or Gel'fand–Graev. We then derive an explicit Singular Value Decomposition for the geodesic X-ray transform. Computations dictate a specific choice of weighted $L^2-L^2$ setting which is equivalent to the $L^2(M, \operatorname{dVol}_\kappa)\to L^2(\partial_+SM,d\Sigma^2)$ one for any $\kappa\in (-1,1)$.

中文翻译：

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等曲率圆盘上测地线X射线变换的范围表征和奇异值分解

对于单位圆盘 $M$ 上恒定曲率的简单度量的单参数族（$4\kappa$ for $\kappa\in (-1,1)$），我们首先明确表示的 Pestov-Uhlmann 范围表征测地线 X 射线变换，通过构建构成其范围和共核的函数的基础。这种范围特征也转化为矩条件 $à$ $la$ Helgason-Ludwig 或 Gel'fand-Graev。然后，我们为测地线 X 射线变换推导出显式奇异值分解。计算决定了加权 $L^2-L^2$ 设置的特定选择，它相当于 $L^2(M, \operatorname{dVol}_\kappa)\to L^2(\partial_+SM,d \Sigma^2)$ 为任何 $\kappa\in (-1,1)$ 一个。