齐次模型集和窗口协方差图
Homometric model sets and window covariograms
摘要：
当两个Delone集共享相同的自相关或Patterson测度时，它们被称为homometric。在给定的切割和投影方案内的模型集LAMBDA是通过内部空间中的窗口W定义的Delone集。LAMBDA的自相关测度是一个纯点测度，其系数可以通过所谓的W的协方差函数来计算。因此，具有相同协方差函数的两个窗口会产生齐次模型集。另一方面，从其衍射图像确定LAMBDA的逆问题最终相当于从其协方差函数重建W。这也被称为Matheron的协方差问题。它在凸几何中得到了很好的研究，近年来得到了一些唯一性结果。然而，对于非凸窗口，唯一性失败的相关方式，使有趣的应用程序的homometry问题出现。我们在一个简单的设置中讨论这一点，并显示了一个平面的例子，不同的homometric模型集。
英文摘要：
Two Delone sets are called homometric when they share the same autocorrelation or Patterson measure. A model set LAMBDA within a given cut and project scheme is a Delone set that is defined through a window W in internal space. The autocorrelation measure of LAMBDA is a pure point measure whose coefficients can be calculated via the so-called covariogram of W. Two windows with the same covariogram thus result in homometric model sets. On the other hand, the inverse problem of determining LAMBDA from its diffraction image ultimately amounts to reconstructing W from its covariogram. This is also known as Matheron's covariogram problem. It is well studied in convex geometry, where certain uniqueness results have been obtained in recent years. However, for non-convex windows, uniqueness fails in a relevant way, so that interesting applications to the homometry problem emerge. We discuss this in a simple setting and show a planar example of distinct homometric model sets.
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