126 problems

Let ${\rm {\mathcal{D}}}$ denote the classical Dirichlet space, let $f\in {\rm {\mathcal{D}}}$ be outer, and let $\mathcal{Z}(f)$ be its boundary zero set. Writing $c_m$ for the ca…

Boundary characterization conjecture. The boundary $\partial\mathcal{L}_\alpha$ coincides with the set

For a linear function space $\mathbb{X}(\mathbb{R}^n)\subset C(\mathbb{R}^n)$, a Finiteness Principle asserts that there are $k\in\mathbb{N}$ and $C>0$ such that, for every finite…

Let $N$ be the Nevanlinna class of holomorphic functions on the unit disk with bounded characteristic, namely those satisfying … where $\log^+x=\max\{0,\log x\}$. Bloch–Nevanlinna…

Let $L(p,\theta,q)$ denote the spaces introduced in the preceding interpolation theorem, and let $[\,\ ]_\sigma$ denote the complex interpolation functor. For $0<p_0<1$ and…

Let $\{x(n)\}$ be a multivariate stationary process with spectral density $W$ on the unit circle $\mathbb{T}$. A matrix function $\Phi$ is Hermitian when $\Phi=\Phi^{*}$, and…

The spaces $A^p_\alpha$ and $\Lambda_\alpha$ are function spaces on the unit ball ${{\mathbb B}_n}$, with $p$ denoting the integrability exponent. The paper refers to a preceding r…

General surjectivity conjecture. Some much more general surjectivity result should hold for affine systems generated by $\psi$, allowing synthesis surjectively onto $L^p$, Hardy, o…

Let $C^{k,\omega }({\bf R}^{n})$ be the space of functions on ${\bf R}^{n}$ with $k$ derivatives whose $k$th-order derivatives have modulus of continuity controlled by $\omega$. It…

Theorem concerns characterizations of alpha-modulation spaces, with parameter $\alpha$ and exponent $p$; the relevant inhomogeneous Besov space is denoted by…

Let $C_p(X)$ denote the space of continuous real-valued functions on a set of reals $X$, with the topology of pointwise convergence. Let $\mathfrak{rez}$ be the least cardinality o…

For $p=0,1$, let the associated metric be the BKM metric, and let $L^{{\rm exp}}$ and $L^{x{\rm log}x}$ denote the Zygmund spaces appearing in the construction under discussion. BK…

Uncomplemented Hilbert-subspace conjecture. For every $p\in[1,2)$, the space $L_p$ contains an uncomplemented subspace $W$ isomorphic to $l_2$.

Let $X$ and $L$ be continuous domains, and let $T$ denote the domain used in the Scott function spaces $[X\to T]$ and $[T\to L]$. The paper considers finite-layer truncation maps a…

Let ${\mathcal{B}}_L$ denote the unit ball at depth $L$ in the univariate ReLU setting. Depth-three stabilization conjecture. The unit balls stabilize after depth $L=3$, namely … C…

Let $\Phi$ be the function considered above, and let $\mathcal{BV}_p(0,2)$ denote the space of functions of bounded $p$-variation on $(0,2)$. The preceding result gives…

Let $\mathbb{D}^2$ denote the bidisc, and let $\mathfrak{D}(\mathbb{D}^2)=\mathfrak{D}_{(1,1)}(\mathbb{D}^2)$ be the bidisc Dirichlet space. Let $\varphi=\frac{\widetilde{p}}{p}$ b…

Let $\mathcal{V}^{p,alpha}_k$ be the vanishing Campanato space consisting of functions satisfying … It contains the closure of smooth functions in $\mathcal{L}^{p,alpha}_k$. Vanish…

$L^p$-boundedness conjecture. The function $\widetilde{H}$ should resolve the issue of $L^p$-unboundedness: unlike $\widetilde{G}$ for $p>2$ and $H$ for $1<p<2$, it is expected to…

Triebel's composition conjecture. If $s>1+\frac{1}{p}$, this map is bounded if and only if

Tight fitting conjecture. (a) The embedding $A^p_\alpha\subset A^q_\beta$ is a tight fitting if and only if

Let $v$ be a weight on the unit disk and let $p$ be the parameter appearing in Proposition 1. Assume that the function $x\mapsto v(1-x)x^{\frac{1}{p}-\epsilon}$ is almost increasin…

Let $k\in\mathbb N$, and consider approximation by shallow ReLU$^\alpha$-networks in the space $C^{k-1}$. Approximation conjecture. A similar analysis may apply to approximation in…

Let $A^{s,\tau}_{p,q}(\mathbb{R}^n)$ denote either a Besov-type or Triebel--Lizorkin-type space, with $s>0$, $p,q\in[1,\infty]$ (and $p<\infty$ in the Triebel--Lizorkin case), and…

Gribonval–Hardin–Nielsen conjecture. For unconditional bases, one has