Universal discretization

TL;DR

This paper introduces universal point sets for discretizing integral norms of multivariate trigonometric polynomial subspaces, leveraging (t,r,d)-nets to achieve optimal order accuracy across various subspaces.

Contribution

It constructs universal discretization sets for multivariate trigonometric polynomial subspaces using (t,r,d)-nets, providing a unified approach for integral norm approximation.

Findings

Universal sets are optimal for discretizing integral norms.

Construction is based on deep results on (t,r,d)-nets.

Sets work uniformly across a collection of subspaces.

Abstract

The paper is devoted to discretization of integral norms of functions from a given collection of finite dimensional subspaces. For natural collections of subspaces of the multivariate trigonometric polynomials we construct sets of points, which are optimally (in the sense of order) good for each subspace of a collection from the point of view of the integral norm discretization. We call such sets universal. Our construction of the universal sets is based on deep results on existence of special nets, known as (t,r,d)-nets.