Homometric model sets and window covariograms
Homometric model sets and window covariograms
复制标题
均质模型集和窗口协方差图
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
U. Grimm
中科院分区：
文献类型：
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作者：
M. Baake;U. Grimm
当两个Delone集共享相同的自相关或Patterson度量时，称为同构。给定切割和投影方案内的模型集Λ是通过内部空间中的窗口W定义的DELONE集。Λ的自相关度量是一个纯点度量，其系数可以通过具有相同协方差函数的两个窗口的所谓协方差函数来计算，从而得到同构模型集。另一方面，从其衍射图像确定Λ的反问题最终相当于从其协方差函数重建W。这也被称为马瑟龙协方差函数问题。它在凸几何中得到了很好的研究，近年来也得到了一定的唯一性结果。然而，对于非凸窗，唯一性以一种相关的方式失效，因此在同构问题中出现了有趣的应用。我们在一个简单的环境中讨论这一点，并展示了一个不同的同构模型集的平面例子。
Two Delone sets are called homometric when they share the same autocorrelation or Patterson measure. A model set Λ 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 Λ 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 Λ 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.