标题： 西格尔-拉东变换的横截动力系统 显示英文标题

标题： Siegel-Radon transforms of transverse dynamical systems

摘要： 我们扩展了Helgason关于广义Radon变换的经典定义，该定义适用于局部紧致空间群$G$的一对齐次空间，将其推广到一个更广泛的框架中，在这个框架中其中一个空间被$G$上的一个可能非齐次的动力系统所取代，并且带有合适的横截面。 这一通用框架涵盖了文献中研究的许多例子，包括几何数论中的Siegel（或$\Theta$-）变换和Marklof-Strömbergsson变换，平移曲面上的Siegel-Veech变换，以及时频分析中的Zak变换。 我们的主要应用涉及动力系统$(X, \mu)$，其中横截面是由分离横截面诱导的。我们建立了与之相关的Siegel-Radon变换有界性、可积性和平方可积性的标准，并展示了如何利用这些变换将诱导的$G$表示嵌入到$L^p(X, \mu)$中，对于适当的$p$值来说。 这些结果特别适用于近似格子的包络及其某些“稀疏化”，包括在可解情形下的任意正密度子集。 在切削和投影集的特殊情况下，我们推导出对偶变换的显式公式，并在Heisenberg群的特殊情况下，通过非周期Zak变换，提供了Schrödinger表示在Heisenberg群中正密度子集的近似格点的包络的$L^2$-空间中的等距嵌入。 显示英文摘要

摘要： We extend Helgason's classical definition of a generalized Radon transform, defined for a pair of homogeneous spaces of an lcsc group $G$, to a broader setting in which one of the spaces is replaced by a possibly non-homogeneous dynamical system over $G$ together with a suitable cross section. This general framework encompasses many examples studied in the literature, including Siegel (or $\Theta$-) transforms and Marklof-Str\"ombergsson transforms in the geometry of numbers, Siegel-sVeech transforms for translation surfaces, and Zak transforms in time-frequency analysis. Our main applications concern dynamical systems $(X, \mu)$ in which the cross section is induced from a separated cross section. We establish criteria for the boundedness, integrability, and square-integrability of the associated Siegel-Radon transforms, and show how these transforms can be used to embed induced $G$-representations into $L^p(X, \mu)$ for appropriate values of $p$. These results apply in particular to hulls of approximate lattices and certain "thinnings" thereof, including arbitrary positive density subsets in the amenable case. In the special case of cut-and-project sets, we derive explicit formulas for the dual transforms, and in the special case of the Heisenberg group we provide isometric embedding of Schr\"odinger representations into the $L^2$-space of the hulls of positive density subsets of approximate lattices in the Heisenberg group by means of aperiodic Zak transforms.