96 problems

For an integer $d\geq 2$, let $\mathbb C^d$ be the $d$-dimensional complex vector space. For $x,\omega\in\mathbb Z_d$, define the cyclic translation and modulation operators by … w…

Let a frame be a sequence in a Hilbert space that provides stable expansions, and call it bounded below in norm when its elements have norms uniformly bounded away from zero. Feich…

Let $\mathcal F_{\bm A}$ be the phase-retrieval map associated with a matrix $\bm A\in\mathbb C^{M\times N}$, and let $N_{\min}(\mathbb C^M)$ denote the smallest $N$ for which some…

Let $\mathbf{u}_1,\ldots,\mathbf{u}_m\in\mathbb{C}^{d}$ satisfy $\|\mathbf{u}_i\|\leq 1$ for all $i$, and suppose there are universal constants $\eta\geq 2$ and $\theta>0$ such tha…

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…

Let $S\in\mathcal{M}_d(\mathbb{C})^+$ and let $\mathbf{a}=(a_i)_{i\in\mathbb{I}_k}\in(\mathbb{R}_{>0}^k)^\downarrow$. Define … and, for a strictly convex unitarily invariant norm…

Weak $d\times 2d$ conjecture. For every $d$, there exists a $d\times 2d$ ETF.

Let $H$ be a Hilbert space, let $\mathcal{M}=(m_n)$ be a sequence of scalars, and let $\mathcal{X}={x_n}$ and $\mathcal{Y}={y_n}$ be sequences in $H$. Balazs–Stoeva conjecture. The…

Let $\mathcal{F}^{\mathbf R}_{k,n}$ denote the space of real spherical tight frames with $k$ frame vectors in dimension $n$. Connectedness conjecture. For $k,n\in\mathbf{N}$ with…

Let $D$ be an operator, let $g$ be a vector, and let $\Lambda=\{\lambda_k\}_{k\in\mathbb{N}_0}$ be a sequence of nonnegative integers. Suppose that $\{D^k g\}_{k\in\mathbb{N}_0}$ i…

Aldroubi–Cabrelli–Krishtal–Molter conjecture. This system is not a frame for $H$.

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…

ACKM's conjecture. The system $\{D^\lambda b:\lambda\in\Lambda\}$ is a frame for $cell^2$.

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…

Let $A,P\in\mathrm{GL}(d,\mathbb{R})$ and $\psi\in L^2(\mathbb{R}^d)$. Consider the wavelet system … Assume that this system is an orthonormal basis, or more generally a Parseval f…

Let $H$ be a Hilbert space, let $A$ be a normal operator on $H$, and let $S$ be a finite subset of $H$. For every $n\geq 0$ and $x\in S$ such that $A^n x\ne 0$, consider the normal…

Let $A$ be a bounded normal operator on an infinite-dimensional Hilbert space $\mathcal H$, and let $\mathcal G\subset\mathcal H$. Normalized-orbit non-frame conjecture. The system…

Let $A$ be an operator and let $\mathcal G$ be a countable subset of a Hilbert space such that the conclusion of Theorem … remains true when $A$ is normal and reductive. The preced…

Müntz–Szász redundancy conjecture. The system $\\{D^\lambda b: \lambda\in\Lambda\\}$ is a frame for $\ell^2(\mathbb N)$.

All-dimensional ETF existence conjecture. Such equiangular tight frames exist for every $d$.

Splittability conjecture. All full-spark frames are $s$-splittable with sufficiently small $\beta>0$.

Let $\{T^k\varphi\}_{k=0}^\infty$ be a Carleson frame as in Theorem 19408b, with $\{\lambda_k\}_{k=1}^\infty\subset\mathbb{D}$ positive and strictly increasing. For…

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