The HRT conjecture on linear independence of time-frequency shifts

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The HRT conjecture on linear independence of time-frequency shifts

   Let $g$ be a nonzero element of $L^2(\mathbb{R})$ and let $\Lambda=\{(a_k,b_k)\}_{k=1}^N\subset\mathbb{R}^2$. Define the finite set of time-frequency shifts

   $$\mathcal{G}(g,\Lambda)=\{e^{2\pi i b_k\cdot}g(\cdot-a_k):k=1,2,\ldots,N\}.$$

   HRT conjecture. The set $\mathcal{G}(g,\Lambda)$ is linearly independent in $L^2(\mathbb{R})$. This open problem concerns the basic linear independence of finite collections of time-frequency shifts and is relevant to the invertibility and numerical conditioning of finite Gabor systems. It was posed in 1990 by C. Heil, J. Ramanathan, and P. Topiwala and is known as the HRT conjecture.

   source: Kasso A. Okoudjou, “An invitation to Gabor analysis”, arXiv:1812.08647 (2019).

2. HRT conjecture on linear independence of time-frequency shifts

   Let $d\geq 1$, let $g\in L^2(\mathbb{R}^d)\setminus\{0\}$, and let $\Lambda=\{z_1,\ldots,z_N\}\subset\mathbb{R}^{2d}$ be a finite set of distinct points. For $z=(x,\omega)\in\mathbb{R}^{2d}$, define the time-frequency shift $\pi(z)g=M_\omega T_xg$, where

   $$T_xg(t)=g(t-x),\qquad M_\omega g(t)=e^{2\pi i t\cdot\omega}g(t).$$

   Write $G(g,\Lambda)=\{\pi(z_k)g\}_{k=1}^N$. HRT conjecture. The set $G(g,\Lambda)$ is linearly independent in $L^2(\mathbb{R}^d)$. This is a central open problem in Gabor analysis concerning finite collections of time-frequency shifts of a nonzero square-integrable function.

   source: Fabio Nicola and S. Ivan Trapasso, “A note on the HRT conjecture and a new uncertainty principle for the short-time Fourier transform”, arXiv:1911.12241 (2019).