  We construct a least squares approximation method for the recovery of
complex-valued functions from a reproducing kernel Hilbert space on $D \subset
\mathbb{R}^d$. The nodes are drawn at random for the whole class of functions
and the error is measured in $L_2(D,\varrho_D)$. We prove worst-case recovery
guarantees by explicitly controlling all the involved constants. This leads to
new preasymptotic recovery bounds with high probability for the error of
Hyperbolic Fourier Regression on multivariate data. In addition, we further
investigate its counterpart Hyperbolic Wavelet Regression also based on
least-squares to recover non-periodic functions from random samples. Finally,
we reconsider the analysis of a cubature method based on plain random points
with optimal weights and reveal near-optimal worst-case error bounds with high
probability. It turns out that this simple method can compete with the
quasi-Monte Carlo methods in the literature which are based on lattices and
digital nets.
