Universal discretization
Provides a universal discretization approach for integral norms in multivariate trigonometric polynomial subspaces, addressing a fundamental problem in approximation theory.
The paper constructs point sets that are optimally good for discretizing integral norms across a collection of subspaces of multivariate trigonometric polynomials, achieving optimal order for each subspace. The construction relies on deep results about (t,r,d)-nets.
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.