14 problems

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

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…

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…

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

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,…

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

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…

Let $\beta_m$ be the level-$m$ vertex set of the Sierpinski gasket, let $y$ be a non-boundary vertex in $\beta_m$, and let $\phi_y^{(m)}$ be the pointwise sampling function satisfy…

Let $\psi_m$ be a sampling function on level $m$ of the Sierpinski gasket $SG$. L2 decay conjecture. There is a constant $c$, numerically around $1.1$, such that … Unlike the corre…

Let $\psi_m$ denote a sampling function on level $m$ of the Sierpinski gasket. Uniform boundedness conjecture. The sampling functions are uniformly bounded by some constant, numeri…