27 problems

Let $\Lambda$ range over periodic configurations of arbitrary fixed density in phase space, and consider the associated quantum packing, covering, and paving problems. The hexagona…

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…

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…

Random-index-set singular-value conjecture. A version of Theorem should hold when $|\Lambda|=O(M)$; in particular, there should exist constants $0<k<K$ independent of $M$ such that…

Define the B-splines by $Q_1(x):=\chi_{[-1/2,1/2]}(x)$ and $Q_{n+1}(x):=(Q_n*Q_1)(x)$ for $n\geq 1$. Let $\mathcal{F}(Q_n)$ be the set of $(a,b)\in\mathbb{R}_+^2$ for which…

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

Luef's conjecture. If such a lattice generates a Gabor frame, then its volume cannot exceed

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, $\alpha\to1^+$,

Conjecture A. If $(\Omega,\Lambda)$ is a transcendental pair, then $(g_\Omega,\Lambda)$ is a Gabor frame for $L^2(\mathbb R^n)$ if and only if

Higher-dimensional multiplier conjecture. If $|\det(AB)|\leq 1$, then $h\in L^{\infty}(\mathbf{R}^d)$ is a functional Gabor frame multiplier if and only if $h$ is unimodular and

Uniform erasure-robustness conjecture. Suppose that $|\Lambda|=O(M\log^\alpha M)$, where $\alpha\geq0$ is to be specified. Then, for every $p\in(0,1)$, $\Delta(p)\geq C$ with high…

Let $g$ be a Steinhaus window, meaning that $g(j)=\frac{1}{\sqrt{M}}e^{2\pi i y_j}$ for $j\in\mathbb{Z}_M$, where the $y_j$ are independent and uniformly distributed on $[0,1)$. Le…

Let $a,b>1$ be arbitrary positive real numbers. Consider the three assertions: (i) $\log_b a\leq 1$; (ii) there exists $\psi\in L^2(\mathbb R_+)$ such that ${\cal MD}(\psi,a,b)$ is…

Daubechies–Gröchenig conjecture. The system $\mathcal{G}(g,\alpha\mathbb{Z}\times\beta\mathbb{Z})$ is a Gabor frame for $L^2(\mathbb{R})$ if and only if

Let $g$ be a Gaussian window and let $\Lambda$ range over lattices with fixed redundancy. Denote by $A$ and $B$ the lower and upper frame bounds of the associated Gabor frame. Simu…

Frame-stability conjecture. For more general Hamiltonians, the evolution $\mathcal{G}(g,\Lambda)\mapsto\mathcal{G}_t(g,\Lambda)$ preserves the frame property. For quadratic Hamilto…

Non-frame conjecture. The Gabor system $\mathcal{G}(B_2,a_0,b_0)$ is not a frame. Furthermore, it is not a frame along the hyperbolas

Frame set conjecture. The frame set of $B_n$ is the maximal set of parameters $(a,b)$ that avoids the known obstructions.

Let $f$ be a nonzero function in $L^2(\mathbb{R})$. A time-frequency translate of $f$ is a function of the form $M_\omega T_xf$, where $x,\omega\in\mathbb{R}$, $T_xf(t)=f(t-x)$, an…

Modified conjecture. Every continuous totally positive function satisfies

Daubechies' conjecture. The frame set is

Let $(\Omega_r)_{r\in\mathcal{I}}$ be a covering of the phase space $\mathbb{R}^{2d}$, and assign to each member of the covering a frame from a family of Gabor frames. The resultin…

A frame is localized with respect to a reference frame, and its associated frame density is obtained from the density of the localization index set in a discrete abelian group, aft…