On the linear independence of spikes and sines

TL;DR

This paper surveys the linear independence of spikes and sines, providing new probabilistic results on spectral norms of random Fourier submatrices using advanced extrapolation techniques.

Contribution

It introduces new results on the linear independence of randomly chosen spikes and sines, connecting to spectral norms of random Fourier submatrices.

Findings

New probabilistic bounds for spectral norms

Results applicable to random spike and sine configurations

Enhanced understanding of Fourier matrix substructure

Abstract

The purpose of this work is to survey what is known about the linear independence of spikes and sines. The paper provides new results for the case where the locations of the spikes and the frequencies of the sines are chosen at random. This problem is equivalent to studying the spectral norm of a random submatrix drawn from the discrete Fourier transform matrix. The proof involves depends on an extrapolation argument of Bourgain and Tzafriri.