72 problems

Heil–Ramanathan–Topiwala conjecture. For every nonzero $g\in L^2(\mathbb{R})$ and every finite set of distinct points $\Lambda=\{(a_k,b_k)\}_{k=1}^N\subset\mathbb{R}^2$, the system…

Let $\varphi(t)=2^{1/4}e^{-\pi t^2}$ be the Gaussian window, and let $M_k$ and $T_n$ denote modulation and translation by $k,n\in\mathbb Z$, respectively. For every…

HRT conjecture. The functions

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…

Linear-independence conjecture. If $f\ne 0$, then $S(f,\Lambda)$ is linearly independent over $\mathbb{C}$. This problem is motivated by Gabor analysis; the source gives no evidenc…

Let $1<p,q<\infty$ satisfy … For $f\in L^2(\mathbb R)$, let $\mathcal G(f,1,1)$ denote the associated Gabor system, and let $H^s(\mathbb R)$ be the usual $L^2$-Sobolev space. Bened…

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 $\alpha,\beta>0$ be the time- and frequency-shift parameters of a Gabor system, and write $\theta=\alpha\beta$. Suppose that the canonical dual atom has the same quality as the…

Let $0<\eta\leq\frac{1}{3}$ and let … for $(a_0,a_1,\ldots,a_N)\in\mathbb{C}^{N+1}$. The preceding theorem states that, modulo a trivial transformation, every solution of the ambig…

Bueckner's conjecture. If $u$ is a Hermite function, then every ambiguity partner of $u$ is trivial.

Let $f$ and $g$ be functions or distributions for which the short-time Fourier transform $\operatorname{STFT}_g f$ is defined, and let $L^p_v$ and $M^p_v$ denote the weighted Lebes…

Let $g$ be a Schwartz function and let ${\mathcal G}(g,\alpha{\mathbb Z}\times\beta{\mathbb Z})$ be a Gabor frame, where $\alpha$ and $\beta$ are lattice constants. Janssen's conje…

A locally compact abelian group is a locally compact topological group whose group operation is commutative. Let $L^2(\G)$ denote the associated Hilbert space of square-integrable…

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…

FAB95's optimality conjecture. This value is optimal among all pulse shapes that create an OFDM system with spectral efficiency $1/2$. The claim concerns the best possible time-fre…

For $g in L^2(\mathbb{R}^d)$ and $\lambda=(x,\omega)\in\mathbb{R}^{2d}$, define the time-frequency shift … For a lattice $\Lambda\subseteq\mathbb{R}^{2d}$, let $D(\Lambda)$ denote…

Let $\mathcal{G}_1(a,b)=\{M_{b\ell}T_{ak}h_1\mid(k,\ell)\in\mathbb{Z}^2\}$ be the Gabor system over the rectangular lattice $a\mathbb{Z}\times b\mathbb{Z}$, where … with…

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…

DGL's conjecture. The two displayed conditions are equivalent when

Sleep spindle annotation and the exploration of spindle dynamics use the synchrosqueezing transform (SST) together with the proposed bootstrap-based uncertainty quantification. Unc…

Sleep spindle durations and instantaneous-frequency dynamics are difficult to estimate from electroencephalogram signals because of their thin-tail structure and background noise.…

Let $T_{I,J}$ be the operator defined by … Fix $\epsilon>0$, let $I=[-R,R]$ and $J=[-T,T]$, and consider the eigenvalues of $T_{I,J}$. Near-one eigenvalue conjecture. The number of…

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…