23 problems

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 opera…

Let $\Delta\subset\mathbb{R}^{2}$ be a measurable set of finite measure, let $A=|\Delta|$, let $\varphi=h_{0}$ be the Gaussian, and let $V_{\varphi}f$ denote the short-time Fourier…

Let $G$ be a finite Abelian group, let $V_g f$ denote the time-frequency representation appearing in the paper, let $\widehat f$ denote the Fourier transform of $f$, and write…

Let $R_{g,\Lambda}$ denote the matrix associated with the Gaussian $g$ and a lattice $\Lambda$, and let ${x_1}(\Lambda)$ denote the lattice parameter used in the source. Minimal co…

HRT conjecture. The functions

HRT conjecture. The functions $g_k$, $1\le k\le N$, are linearly independent in $L^2(\mathbb R,\mathrm{d}x)$. This is a central open problem in time-frequency analysis, also called…

Let $\Lambda=\{(x_k,y_k):1\leq k\leq N\}\subset \mathbb{R}^d\times\mathbb{R}^d$ be a finite set of distinct points, and let $0\neq f\in L^2(\mathbb{R}^d)$. HRT conjecture. The fini…

Universality conjecture. For every such $\Omega$,

Let $g(t)=\exp(t-e^t)$, and let $\gamma(x,\xi)=\exp(-a|x|-b|\xi|)$ with $a,b>0$. Suppose that $f\in L^2(\mathbb{R})$ has exponential time-frequency concentration, meaning … Let…

Let $Q_2=Q_1*Q_1$, where $Q_1(x):=\chi_{[-1/2,1/2]}(x)$, and let $\mathcal{G}(Q_2,a,b)$ denote the associated Gabor system. Nielsen's conjecture. The Gabor system…

Lemvig and Nielsen's conjecture. The Gabor system $\mathcal{G}(Q_2,a,b)$ is not a frame along the hyperbolas

Universal Gaussian Gabor asymptotic conjecture. There exists a constant $C_\Lambda$, depending on the geometry of $\Lambda(1)$, such that

Strohmer's conjecture. As the critical density is approached from above, $\alpha\to1^+$,

Let $\Lambda\subset\mathbb{R}^{2}$ be a fixed finite set, let $f\in L^{2}(\mathbb{R})$ be nonzero, and write … Here $\mathcal{G}(f,\Lambda)$ is the associated finite Gabor system.…

Let $g_0(t)=2^{1/4}e^{-\pi t^2}$ be the Gaussian, let $\Lambda\subset\mathbb{R}^2$ be a lattice of fixed density $\delta$, and let $A$ and $B$ denote the sharp lower and upper fram…

Strohmer and Beaver's conjecture. Among Gaussian Gabor systems with fixed lattice density greater than $1$, the condition number $\operatorname{cond}(S_{g_0,\Lambda})=B/A$ is minim…

Let $g\in S_0({\mathbb R})$ and let $\Lambda \subset {\mathbb R}^2$ be a lattice such that $(g,\Lambda)$ is a Riesz basis for its closed linear span $\mathcal G(g,\Lambda)$. Time-f…

Let $f\in L^2(\operatorname{\mathbb{R}})$ be nonzero, and let $\Lambda\subset\operatorname{\mathbb{R}}^2$ be a finite set. For $\lambda=(x,\omega)\in\operatorname{\mathbb{R}}^2$, w…

Rotational invariance conjecture. The following are equivalent:

Heil's four-point conjecture. The system $\mathcal{G}(g,\Lambda)$ is linearly independent.

Schwartz-function sub-conjecture. For every nonzero $g\in S(\mathbb{R})$ and every such $\Lambda$, the system $\mathcal{G}(g,\Lambda)$ is linearly independent in $L^2(\mathbb{R})$.

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…

Let $(t_j,\xi_j)_{j=1}^n$ be $n\ge 2$ distinct points in the plane, and let $f:\mathbb R\to\mathbb C$ be a Schwartz function. A nontrivial linear dependence means that not all coef…