This paper is a contemporary review of QMC ('quasi-Monte Carlo') methods, that is, equal-
weight rules for the approximate evaluation of high-dimensional integrals over the unit cube …

In this paper, we present algorithms for the approximation of multivariate periodic functions
by trigonometric polynomials. The approximation is based on sampling of multivariate …

In this paper, we suggest approximate algorithms for the reconstruction of sparse high-
dimensional trigonometric polynomials, where the support in frequency domain is unknown …

Riemann theta functions play a crucial role in the field of nonlinear Fourier analysis, where
they are used to realize inverse nonlinear Fourier transforms for periodic signals. The …

Sparse grid discretisations allow for a severe decrease in the number of degrees of freedom
for high-dimensional problems. Recently, the corresponding hyperbolic cross fast Fourier …

We consider the approximate recovery of multivariate periodic functions from a discrete set
of function values taken on a rank-1 lattice. Moreover, the main result is the fact that any (non …

A comprehensive overview of lattice rules and polynomial lattice rules is given for function
spaces based on ℓ𝑝 seminorms. Good lattice rules and polynomial lattice rules are defined …

We consider rank-$1 $ lattices for integration and reconstruction of functions with series
expansion supported on a finite index set. We explore the connection between the periodic …

The approximation of problems in dd spatial dimensions by trigonometric polynomials
supported on known more or less sparse frequency index sets I ⊂ Z^ d I⊂ Z d is an …

With given Fourier coefficients the evaluation of multivariate trigonometric polynomials at the
nodes of a rank-1 lattice leads to a one-dimensional discrete Fourier transform. In many …