Let (t_j,xi_j)_j=1^n be n≥ 2 distinct points in the plane. A time-frequency translate of a function f∈ L^2( R) is a function of the form f(x+t_j)e^2 pi i xi_jx. HRT conjecture. For any such distinct points, there is no nontrivial f∈ L^2( R) and no nontrivial coefficients d_1,…,d_n satisfying ∑_j=1^n d_j f(x+t_j)e^2 pi i xi_jx=0 for almost every x∈ R. The conjecture asserts linear independence of every finite collection of distinct time-frequency translates; the paper proves it for four points in a (2,2) configuration, while the general case remains open.

HRT conjecture on linear independence of time-frequency translates

Let $(t_j,\xi_j)_{j=1}^n$ be $n\ge 2$ distinct points in the plane. A time-frequency translate of a function $f\in L^2(\mathbb R)$ is a function of the form $f(x+t_j)e^{2\pi i\xi_jx}$ . HRT conjecture. For any such distinct points, there is no nontrivial $f\in L^2(\mathbb R)$ and no nontrivial coefficients $d_1,\ldots,d_n$ satisfying

\sum_{j=1}^n d_j f(x+t_j)e^{2\pi i\xi_jx}=0

for almost every $x\in\mathbb R$ . The conjecture asserts linear independence of every finite collection of distinct time-frequency translates; the paper proves it for four points in a $(2,2)$ configuration, while the general case remains open.

Equivalent formulations 1

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 translates
Let $(t_j,cxi_j)_{j=1}^n$ be $n\ge 2$ distinct points in the plane, and let $f:\mathbb R\to\mathbb C$ be an $L^2$ function. A nontrivial linear dependence means that not all coefficients $d_i$ vanish and
$\sum_{j=1}^n d_i f(x+t_j)e^{2\pi i\xi_jx}=0$
for almost every $x\in\mathbb R$ . HRT conjecture. There is no nontrivial $L^2$ function satisfying such a nontrivial linear dependence. The conjecture is known for points lying on a lattice, in particular for any three points, and when all points are collinear; the general case remains open.
source: Ciprian Demeter, “Linear independence of time frequency translates for special configurations”, arXiv:1006.0732 (2016).

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Primary source

Ciprian Demeter and Alexandru Zaharescu, “Proof of the HRT conjecture for (2,2) configurations”, arXiv:1006.0735 (2010).

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HRT conjecture on linear independence of time-frequency translates

Equivalent formulations 1

The HRT conjecture on linear independence of time-frequency translates

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