39 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…

Gröchenig's conjecture. For every $n\in\mathbb{N}$,

Consider the problem of finding an optimal lattice sampling pattern for Gaussian Gabor systems. Let a lattice mean a discrete subgroup of the relevant phase space, and compare latt…

Let $f\in L^2(\mathbb{R})$ be nonzero, and let $\{(x_j,\omega_j)\}_{j=1}^N$ be a finite collection of time-frequency shifts. The corresponding shifted functions are the functions…

Let $\mathcal{G}_N(\phi_i,\Lambda_i)$ be a Gabor system in $L^2(\mathbb{R}^d)$, where the disjoint union $\bigcup_{i=1}^N\Lambda_i$ is formed from the parameter sets…

Let $\mathbf{G}(g,T,S)$ be a Gabor system that is an orthonormal basis in $L^2(\mathbb{R}^d)$, with window function $g$ of compact support. A measurable set…

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…

Twisted-group-ring injectivity conjecture. If $0\ne\theta\in\mathbb{C}*\mathcal{G}_n$ and $0\ne f\in L^2(\mathbb{R}^n)$, then $\theta f\ne0$. This is the operator-theoretic formula…

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 $\alpha,\beta>0$ 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 $g$ 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…

For $(m,n)\in\mathbb{Z}^2$, let $\gamma_{m,n,0,0}$ denote the biorthogonal element associated with the Gaussian Gabor atom indexed by $(m,n)$, and let $c_{m,n}>0$. The Gaussian Gab…

Let $\mathcal{F}$ be a family in a separable Hilbert space $\mathcal{H}$. The family is complete if its closed linear span is $\mathcal{H}$, and it is a weighted lower semi frame i…

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…

Let $g$ be the Gaussian window and let $a,b>0$. Consider the Gabor system $G(g,a,b)$ on the rectangular lattice $a\mathbb{Z}\times b\mathbb{Z}$. Daubechies' Gaussian Gabor frame co…

Rathmair's conjecture. The local stability and Cheeger constants of $f$ should be finite and non-zero.

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

Let $\varphi(t)=2^{1/4}e^{-\pi t^2}$ be the Gaussian atom, and let $\alpha,\beta>0$. For the rectangular lattice $\alpha\mathbb{Z}\times\beta\mathbb{Z}$, write…

The frame set conjecture for Hermite functions. The frame set for Hermite functions of even orders is

Gröchenig's conjecture. If $phi$ is totally positive and is not the one-sided exponential, then

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 $\phi_n$ be a twisted $B$-spline of order $n$, and let $T_{(k,l)}^t$ denote the twisted translation indexed by $(k,l)\in\mathbb{Z}^2$. Riesz-sequence conjecture. The system … i…

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…