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It has been observed by several authors that well-known periodizationstrategies like tent or Chebychev transforms lead to remarkable results for therecovery of multivariate functions from few samples. So far, theoreticalguarantees are missing. The goal of this paper is twofold. On the one hand, wegive such guarantees and briefly describe the difficulties of the involvedproof. On the other hand, we combine these periodization strategies with recentnovel constructive methods for the efficient subsampling of finite frames in$\mathbb{C}$. As a result we are able to reconstruct non-periodic multivariatefunctions from very few samples. The used sampling nodes are the result of atwo-step procedure. Firstly, a random draw with respect to the Chebychevmeasure provides an initial node set. A further sparsification techniqueselects a significantly smaller subset of these nodes with equal approximationproperties. This set of sampling nodes scales linearly in the dimension of thesubspace on which we project and works universally for the whole class offunctions. The method is based on principles developed by Batson, Spielman, andSrivastava and can be numerically implemented. Samples on these nodes are thenused in a (plain) least-squares sampling recovery step on a suitable hyperboliccross subspace of functions resulting in a near-optimal behavior of thesampling error. Numerical experiments indicate the applicability of ourresults.

Efficient recovery of non-periodic multivariate functions from few scattered samples