# Letter to the Editor: Linear Independence of Time-Frequency Shifts Up To Extreme Dilations

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## Abstract

Given \(f \in C_0({{\,\mathrm{\mathbb {R}}\,}}^n)\) and a finite set \(\Lambda \subset {{\,\mathrm{\mathbb {R}}\,}}^{2n}\) 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 \(f \in C_0({{\,\mathrm{\mathbb {R}}\,}}^n)\) and finite \(\Lambda \subset {{\,\mathrm{\mathbb {R}}\,}}^{2n}\) 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.

## Keywords

HRT conjecture Time-frequency analysis Short-time Fourier transform## Mathematics Subject Classification

Primary: 42C15 Secondary: 42C40## Notes

### Acknowledgements

The author thanks Radu Balan and Kasso Okoudjou for introducing him to the HRT Conjecture and for helpful discussions about this work. The author also thanks an anonymous reviewer for encouraging him to expand the results contained in earlier drafts.

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