12 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…

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…

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…

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 $g_0(t)=2^{1/4}e^{-\pi t^2}$ be the standard Gaussian, let $\delta>1$, and define … For each separable lattice in this class, let $A$ and $B$ denote the lower and upper frame b…

Let $g_0(t)=2^{1/4}e^{-\pi t^2}$ be the standard Gaussian, let $\delta>1$, and define … For each $\Lambda$ in this class, let $A$ and $B$ denote the lower and upper frame bounds of…

Let $K\subset \mathbb R^d$ and let $B$ be a rational matrix with $\det(B)=1$. Let $s$ be the least common multiple of the denominators of the entries $b_{ij}$. Rational-matrix Fugl…