33 problems

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…

Strong divergence conjecture. There exists an $f\in\mathcal{PW}^{1}_{\pi}$ such that

Let $P(a,z)$ be the self-inversive polynomial defined by … Here $a+r\mathbb{Z}$ is a sampling set for the space $V(Q_m)$, with $m>r$, and $P(a,z)$ arises from the factorization…

Let $PW_\Omega(A_p)$ denote a space of variable bandwidth with global bandwidth $\Omega$ and local variation $p(x)$, so that the local bandwidth at $x$ is $c(x)=\Omega/\sqrt{p(x)}$…

Optimal variance conjecture. The parameter specified by … with the Gaussian function … .

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

Cluster-concentrated basis conjecture. There are $\operatorname{dim}_K-1$ eigenvalues of $PQ$ larger than $1/2$, including $\operatorname{dim}_K-(K+1)$ eigenvalues equal to one and…

Interpolatory basis conjecture. There exist basis functions with the interpolatory property for any space of the form

Sharp-gap sampling conjecture. If

Let $f$ be a band-limited function with $2K+1$ non-zero Fourier coefficients, let $n\geq 2K+1$, and sample training data $(x_i,y_i)$ randomly with $x_i\sim U[-\pi,\pi]$ and…

A randomly subsampled frame is a frame obtained by selecting a random subset of rows from a larger frame. Manova distribution conjecture. The eigenvalues of a randomly subsampled f…

Linear sample-complexity conjecture. The existential sample complexity can be upper bounded by

Lower-bound extension conjecture. The lower bound can be extended to hold without the assumption that $s_{\mu,\epsilon}$ grows slower than $1/\epsilon^p$ for some $p<1$.

Let $\mathcal{H}^{\alpha,\beta}(\mathbb{T}^d)$ be the source spaces of functions on the $d$-dimensional torus with generalized mixed smoothness parameters $\alpha$ and $\beta$, and…

Let $V(\phi)$ be a shift-invariant space, let $k$ and $\nu$ be positive integers, and let $\\{x_i:i\in\mathbb{Z}\\}$ be a separated set. For a function $f$, consider the nonuniform…

Let $Q_m$ denote the spline generator and let $V(Q_m)$ be the shift-invariant spline space it generates. A sequence $\\{x_i:i\in\mathbb{Z}\\}$ is separated if its distinct points h…

Let $p$ be a positive parametrizing function, let $A_p$ be the associated Sturm–Liouville operator, and let $PW_{[0,\Omega]}(A_p)$ denote its spectral subspace. For a set $X$, writ…

Aliasing-norm conjecture. The formula in Proposition should hold for the consistent sampling corresponding to the oblique decomposition

Asymptotic conjecture.

Let $\Lambda = \{\lambda_n\}_{n\in\mathbb{Z}}$ be a complete interpolating sequence for $\mathcal{PW}^{2}_{\pi}$. Let $\mathrm{H}_N$ be the sampling-based approximation operators o…

Pointwise operator-norm divergence conjecture. For every $t\in\mathbb{R}$, there exists a stable LTI system $\mathrm{H}:\mathcal{PW}^{1}_{\pi}\to\mathcal{PW}^{1}_{\pi}$ such that,…

Let a signal on the sphere be band-limited at band-limit $L$, and let spherical designs provide quadrature for computing its spherical harmonic transform. The $2L^2$-sample conject…

Let $\{t_k\}_{k\in\mathbb{Z}}\subset\mathbb{R}$ be an ordered complete interpolating sequence for $\mathcal{PW}_{\pi}^{2}$. Logarithmic lower-bound conjecture. There exists a posit…

Let $\{t_k\}_{k\in\mathbb{Z}}\subset\mathbb{R}$ be an ordered complete interpolating sequence for $\mathcal{PW}_{\pi}^{2}$, let $\phi_k$ be as defined in the source, and let…

Let t_k_{k\in\mathbb{Z}}\subset\mathbb{R} be an ordered complete interpolating sequence for $\mathcal{PW}_{\pi}^{2}$, let $\phi_k$ be as defined in the source, and let…