A Proof of the HRT Conjecture for Widely Spaced Sets

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 $\mathcal{G}(D_{r}f,\Lambda )$ 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.

Added results on dilations of functions and functions with singularities

References in corpus (1)

• Linear independence of time-frequency translates of functions with faster than exponential decay