Let n be a positive integer, let g∈ L^2(R^n) be nonzero, and for x,omega∈R^n let T_x and M_omega denote the translation and modulation operators, respectively. A family is linearly independent if every finite subfamily is linearly independent in the classical sense. HRT conjecture. The set M_omegaT_xg_(x,omega)∈R^2n is linearly independent in L^2(R^n) for all nonzero g∈ L^2(R^n). This is equivalent to the question of whether the associated Gabor spaces are fully interpolating, and remains open despite many partial results on linear independence of time-frequency shifts.

The HRT conjecture on linear independence of time-frequency shifts

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Let $n$ be a positive integer, let $g\in L^{2}(\mathbb{R}^{n})$ be nonzero, and for $x,\omega\in\mathbb{R}^{n}$ let $T_x$ and $M_\omega$ denote the translation and modulation operators, respectively. A family is linearly independent if every finite subfamily is linearly independent in the classical sense. HRT conjecture. The set

\left\{M_{\omega}T_{x}g\right\}_{(x,\omega)\in\mathbb{R}^{2n}}

is linearly independent in $L^{2}(\mathbb{R}^{n})$ for all nonzero $g\in L^{2}(\mathbb{R}^{n})$ . This is equivalent to the question of whether the associated Gabor spaces are fully interpolating, and remains open despite many partial results on linear independence of time-frequency shifts.

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Equivalent formulations 2

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

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).
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).

Sources & referencesView supporting material

Primary source

Eirik Berge, “Interpolation in Wavelet Spaces and the HRT-Conjecture”, arXiv:2005.04964 (2020).

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