Sampling discretization of the uniform norm and applications

TL;DR

This paper investigates methods to discretize the uniform norm of functions in finite-dimensional subspaces using polynomially many samples, improving upon the exponential sample size typically required, through weakened inequalities and subspace restrictions.

Contribution

It introduces new approaches for uniform norm discretization that reduce sample size from exponential to polynomial by relaxing bounds or restricting subspaces.

Findings

Polynomial sample size for discretization achieved under certain conditions

Discretization bounds linked to best m-term bilinear approximation

New theoretical connections between sampling and approximation theory

Abstract

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. Previous known results show that for any $N$-dimensional subspace of the space of continuous functions it is sufficient to use $e^{CN}$ sample points for an accurate upper bound for the uniform norm by the discrete norm and that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. In this paper we focus on two types of results, which allow us to obtain good discretization of the uniform norm with polynomial in $N$ number of points. In the first way we weaken the discretization inequality by allowing a bound of the uniform norm by the discrete norm multiplied by an extra factor, which may depend on $N$. In the second way we impose restrictions on the finite dimensional subspace under consideration. In particular, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best $m$-term bilinear approximation of the Dirichlet kernel associated with the given subspace.