Worst-case Recovery Guarantees for Least Squares Approximation Using Random Samples

Abstract We construct a least squares approximation method for the recovery of complex-valued functions from reproducing kernel Hilbert space on $$D \subset \mathbb {R}^d$$ D ⊂ R d . The nodes are drawn at random whole class functions, and error is measured in $$L_2(D,\varrho _{D})$$ L 2 ( , ϱ ) prove worst-case guarantees by explicitly controlling all involved constants. This leads to new preasymptotic bounds with high probability hyperbolic Fourier regression multivariate data. In addition, we further investigate its counterpart wavelet also based recover non-periodic samples. Finally, reconsider analysis cubature plain points optimal weights reveal near-optimal probability. It turns out that this simple can compete quasi-Monte Carlo methods literature which lattices digital nets.