33 problems

Let $F=[g_1,g_2,\dots,g_{n+k}]$ be an $n\times(n+k)$ matrix, and define … Assume that $\|g_j\|\leq1$ for every $j$. Polynomial upper-bound conjecture. There exists a constant…

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…

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…

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

Let $H$ be a Hilbert space, and let category (c) be the class of frames described in Theorem of the source. A frame is normalizable when normalizing each nonzero vector produces a…

Finite-union conjecture. Every bounded below ASF can be partitioned as a finite union of ARSs. The source's statement uses “ARBs,” but the preceding definition introduces ARSs; thi…

Feichtinger conjecture. The sequence $\{\tau_n\}_n$ can be partitioned into a finite union of Riesz sequences for $\mathcal{H}$. This conjecture was proved by Marcus, Spielman and…

Non-Archimedean Zauner conjecture. For every $d\in\mathbb{N}$, there exist such vectors satisfying $\langle\tau_j,\tau_j\rangle=1$ for all $1\leq j\leq d^2$, with $S_\tau$ diagonal…

Let $p$ be a prime, let $d\in\mathbb{N}$, and let $\langle\cdot,\cdot\rangle$ and $\|\cdot\|$ denote the inner product and norm on $\mathbb{Q}_p^d$. The vectors…

For each dimension $m$, let $\rho_1,\rho_2,\rho_3$ denote the three quantities defined in the paper, and let $\rho(m)$ be the associated complex projection constant. Zauner's conje…

Let $F$ be a frame for $\mathbb{F}^n$, let $\tilde{F}$ be a dual frame, and let $(F,\tilde{F})$ be the associated frame pair. Co-equidistribution characterization conjecture. The f…

Let $F$ be a frame for $\mathbb{F}^n$. A dual of $F$ forms an exclusive Grassmannian pair with $F$ if it is the only dual frame satisfying the paper's Grassmannian coherence minimi…

Let $F$ be a frame for $\mathbb{F}^n$, and let $G$ be one of its dual frames. Write $Gr(F,G)$ for their cross Gramian and let $\mu(Gr(F,G))$ denote its maximum off-diagonal magnitu…

Let $\mathcal{A}$ be a unital C-algebra with the invariant basis number property or let $\mathcal{A}$ be a W-algebra. For $d\in\mathbb{N}$, consider unit inner product frames in th…

Let $\mathcal{A}$ be a unital C-algebra, let $d\in\mathbb{N}$, and let $\mathcal{A}^d$ denote the standard Hilbert C-module. A unit inner product frame has…

Let $(\Omega,\mu)$ be a measure space, let $d\in\mathbb{N}$, and let $\{\tau_\alpha\}_{\alpha\in\Omega}$ be a continuous frame for the Hilbert space $\mathbb{C}^d$. The frame is…

Let $\mathcal{H}$ be a left-Hilbert module that is finite-dimensional as a vector space. Let $X$ be a measure space, let $\mathbb{F}$ be the underlying scalar field, and let…

Let $d$ be a positive integer, let $q$ be a prime power, and let $\mathbb{F}_{q^2}$ be the quadratic extension of the finite field with $q$ elements. A unitary geometry on…

Equivalent-frame inner-product conjecture. $F$ and $G$ are equivalent if and only if

Let $p>0$ with $p\notin 2\mathbb N$, and let $\mu$ be a probability measure on $\mathbb S^{d-1}$. Its $p$-frame energy is … A finite discrete measure is a measure supported on fini…

Chen–Gonzales–Goodman–Kang–Okoudjou's conjecture. If $p\in(p_{k-1},p_k]$ for $k=1,2,\ldots,d$, then $L_k^d$ minimizes the $p$-frame potential when $N=d+1$.

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 $N=m+kd$ points be given in $\mathbb{S}^{d-1}$, where $1\leq m<d$ and $d\geq 2$, and let $\mathbf{A}\in\mathbb{R}^{N\times N}$ be their Gram matrix. The repeated orthonormal se…