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：多位学者已观察到，诸如帐篷变换或切比雪夫变换等经典周期化策略，在从少量样本恢复多元函数时效果显著。然而，目前尚缺乏相应的理论保证。本文旨在达成两个目标：一方面，我们提供此类理论保证并简要阐述证明过程中所涉及的难点；另一方面，我们将这些周期化策略与近期提出的有限框架高效子采样新构造方法相结合，应用于$\mathbb{C}$空间。由此，我们能够从极少量样本中重建非周期多元函数。所采用的采样节点通过两步流程生成：首先，基于切比雪夫测度随机抽取初始节点集；随后，通过稀疏化技术从该节点集中选取规模显著缩小且保持近似性质一致的子集。该采样节点集的规模与投影子空间的维度成线性关系，并适用于整个函数类。该方法基于Batson、Spielman和Srivastava提出的原理，可通过数值计算实现。利用这些节点上的样本，我们在适当的双曲交叉函数子空间上执行（普通）最小二乘采样恢复步骤，从而使得采样误差呈现近最优特性。数值实验验证了本文结果的有效性。