Integral norm discretization and related problems

TL;DR

This paper investigates how to approximate integral norms with discrete measures for finite dimensional spaces, focusing on multivariate trigonometric polynomials, and presents new results alongside a survey of existing literature.

Contribution

It introduces new methods for discretizing integral norms and provides a comprehensive survey of related problems, especially for multivariate trigonometric polynomials.

Findings

New discretization bounds for multivariate trigonometric polynomials

Comparison of discrete and integral norms in finite dimensional spaces

Survey of existing discretization techniques

Abstract

The problem of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure is discussed in the paper. The above problem is studied for elements of finite dimensional spaces. Also, discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of the multivariate trigonometric polynomials with frequencies from a finite set with fixed cardinality. Both new results and a survey of known results are presented.