New perspectives on Fourier matrices

Weilin Li

Session: General

Keywords: Fourier matrix, trigonometric polynomial, sparsity

TL;DR: Summary and exposition on new estimates for Fourier matrices

Abstract: We outline new lower bounds for the smallest singular value of univariate and multivariate non-harmonic Fourier matrices. If the node set has sufficiently small local sparsity at an appropriate scale, then the smallest singular value is primarily determined by the local multiscale geometry of the node set. This illustrates an implicit localization phenomenon of the Fourier transform. We highlight some important conceptual and technical advancements that lead to these bounds, such as the use of refined polynomial interpolation methods and sparsity decomposition.

Submission Number: 62