It has been observed by several authors that well-known periodization strategies like tent or Chebychev transforms lead to remarkable results for the recovery of multivariate functions from few samples. So far, theoretical guarantees are missing. The goal of this paper is twofold. On the one hand, we give such guarantees and briefly describe the difficulties of the involved proof. On the other hand, we combine these periodization strategies with recent novel constructive methods for the efficient subsampling of finite frames in $\mathbb{C}$. As a result we are able to reconstruct non-periodic multivariate functions from very few samples. The used sampling nodes are the result of a two-step procedure. Firstly, a random draw with respect to the Chebychev measure provides an initial node set. A further sparsification technique selects a significantly smaller subset of these nodes with equal approximation properties. This set of sampling nodes scales linearly in the dimension of the subspace on which we project and works universally for the whole class of functions. The method is based on principles developed by Batson, Spielman, and Srivastava and can be numerically implemented. Samples on these nodes are then used in a (plain) least-squares sampling recovery step on a suitable hyperbolic cross subspace of functions resulting in a near-optimal behavior of the sampling error. Numerical experiments indicate the applicability of our results.

翻译：多位学者观察到，诸如帐篷变换或切比雪夫变换等成熟的周期化策略，在从少量样本恢复多变量函数时取得了显著成效。然而，目前尚缺乏理论保障。本文旨在实现双重目标：一方面，我们提供此类理论保障，并简要阐述相关证明中的难点；另一方面，我们将这些周期化策略与近期提出的新型构造方法相结合，用于对$\mathbb{C}$中有限框架进行高效子采样。由此，我们得以从极少量样本中重建非周期多变量函数。所使用的采样节点通过两步流程生成：首先，根据切比雪夫测度进行随机抽取，获得初始节点集；随后，通过一种稀疏化技术，从这些节点中筛选出规模显著缩小但保持同等逼近性质的子集。该采样节点集在投影子空间维度上呈线性扩展，且对整类函数具有普适性。该方法基于Batson、Spielman和Srivastava提出的原理，可实现数值计算。在此基础上，利用这些采样节点，在合适的双曲交叉函数子空间上进行（普通）最小二乘采样恢复，使采样误差呈现近最优特性。数值实验验证了本文结果的适用性。