论文标题
单型组的采样和插值的动力学方法
A dynamical approach to sampling and interpolation in unimodular groups
论文作者
论文摘要
我们在本地紧凑的基团中引入了一个covolume的概念，该概念同时概括了晶格的covolume和beurling密度的互惠，以供umenable，uniumentiumbolume。这种covolume的概念自然源于应用于与点集相关的船体动力系统的横向措施理论。使用类固醇技术，我们证明，根据这一新的covolume概念，在单模型基团上繁殖核希尔伯特空间进行了采样和插值的必要条件。这些条件概括了先前已知的密度定理，用于紧凑的多项式生长组，同时涵盖了重要的新示例，特别是由剪切和项目的方案引起的模型集。
We introduce a notion of covolume for point sets in locally compact groups that simultaneously generalizes the covolume of a lattice and the reciprocal of the Beurling density for amenable, unimodular groups. This notion of covolume arises naturally from transverse measure theory applied to the hull dynamical system associated to a point set. Using groupoid techniques, we prove necessary conditions for sampling and interpolation in reproducing kernel Hilbert spaces on unimodular groups in terms of this new notion of covolume. These conditions generalize previously known density theorems for compactly generated groups of polynomial growth, while also covering important new examples, in particular model sets arising from cut-and-project schemes.
