In this article, we consider the problem of approximating a finite set of data (usually huge in applications) by invariant subspaces generated through a small set of smooth functions. The invariance is either by translations under a full-rank lattice or through the action of crystallographic groups. Smoothness is ensured by stipulating that the generators belong to a Paley-Wiener space, that is selected in an optimal way based on the characteristics of the given data. To complete our investigation, we analyze the fundamental role played by the lattice in the process of approximation.

翻译：在本文中，我们考虑通过由少量光滑函数生成的不变子空间来逼近有限数据集（在实际应用中通常规模庞大）的问题。不变性既可以是满秩格点平移下的不变性，也可以是晶体学群作用下的不变性。光滑性通过要求生成函数属于某个Paley-Wiener空间来保证，该空间基于给定数据的特征以最优方式选取。为完成我们的研究，我们分析了格点在该逼近过程中所起的基础性作用。