Abstract
We establish the linear independence of time-frequency translates for functions \(f\) on \(\mathbb {R}^d\) having one-sided decay \(\lim _{x \in H,\ |x|\rightarrow \infty } |f(x)| e^{c|x| \log |x|} = 0\) for all \(c>0\), which do not vanish on an affine half-space \(H \subset \mathbb {R}^d\).
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Notes
The original HRT conjecture was only for \(\mathbb {R}\), but the question is also open for higher dimensions.
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Acknowledgments
The authors would like to thank Dan Freeman for helpful discussions. The first author was partially supported by NSF Grant DMS-1265711. This work was partially supported by a grant from the Simons Foundation (#244953 Speegle).
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Bownik, M., Speegle, D. Linear Independence of Time-Frequency Translates in \(\mathbb {R}^d\) . J Geom Anal 26, 1678–1692 (2016). https://doi.org/10.1007/s12220-015-9604-8
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DOI: https://doi.org/10.1007/s12220-015-9604-8