  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.
