Approximation and Interpolation of Singular Measures by Trigonometric Polynomials

TL;DR

This paper investigates how well general complex measures, including those supported on curves, can be approximated by trigonometric polynomials on the torus using Wasserstein-1 distance, providing bounds and interpolation methods.

Contribution

It establishes sharp bounds for approximation quality and introduces sum of squares polynomial methods for measure interpolation and characteristic function approximation.

Findings

Sharp lower bounds for measure approximation by trigonometric polynomials.

Upper bounds for computable approximations using known moments.

Sum of squares polynomials effectively interpolate measure supports.

Abstract

Complex signed measures of finite total variation are a powerful signal model in many applications. Restricting to the $d$-dimensional torus, finitely supported measures allow for exact recovery if the trigonometric moments up to some order are known. Here, we consider the approximation of general measures, e.g., supported on a curve, by trigonometric polynomials of fixed degree with respect to the Wasserstein-1 distance. We prove sharp lower bounds for their best approximation and (almost) matching upper bounds for effectively computable approximations when the trigonometric moments of the measure are known. A second class of sum of squares polynomials is shown to interpolate the characteristic function on the support of the measure and to converge to zero outside.