A Proof of the HRT Conjecture for Widely Spaced Sets

TL;DR

This paper proves the HRT conjecture for sets with widely spaced points, showing linear independence of certain time-frequency translates for functions with decay or specific singularities.

Contribution

It establishes the linear independence of time-frequency translates for widely spaced sets, extending the HRT conjecture to new classes of functions and configurations.

Findings

Linear independence for widely spaced sets with decay functions

Existence of dilations making sets linearly independent

Linear independence for functions with singularities like cos(t)/|t|

Abstract

Given $f \in C_0(\mathbb{R}^n)$ and $\Lambda \subset \mathbb{R}^{2n}$ a finite set we demonstrate the linear independence of the set of time-frequency translates $\mathcal{G}(f, \Lambda) = \{\pi(\lambda)f\}_{\lambda\in \Lambda}$ when the time coordinates of points in $\Lambda$ are far apart relative to the decay of $f.$ As a corollary, we prove that for any $f \in C_0(\mathbb{R}^n)$ and finite $\Lambda \subset \mathbb{R}^{2n}$ there exist infinitely many dilations $D_r$ such that $\mathcal{G}(D_rf, \Lambda)$ is linearly independent. Furthermore, we prove that $\mathcal{G}(f, \Lambda)$ is linearly independent for functions like $f(t) = \frac{cos(t)}{|t|}$ which have a singularity and are bounded away from any neighborhood of the singularity.