285 problems

Let $N(d,t)$ denote the smallest cardinality of a spherical $t$-design on the $(d-1)$-dimensional sphere, and let $C_d$ be a sufficiently large positive constant depending only on…

Let $I=[a,b]$ and let $\mathbf{x}$ be a node system, with interval maxima … where $I_i(\mathbf{x})=[x_{i-1},x_i]$ for $1\leq i\leq n$ and $I_{n+1}(\mathbf{x})=[x_n,\infty)$. Bernst…

Let $d>1$ be the dimension and let $r:=1+\frac{1}{p}$, where $p$ is the exponent in the relevant $L_p$-norm. For the Lipschitz function class ${\cal L}$ considered in the paper, th…

de Boor's conjecture. The interpolation by polynomial B-splines of degree $d$ at node averages is bounded by a function depending only on $d$, regardless of the nodes themselves.

Let $m\in\mathbb N$ be an arbitrary derivative order, and consider B-spline estimators of a univariate function subject to an $m$th-order nonnegative derivative constraint. The ass…

Schenck–Stiller's "2r+1" conjecture. For degree at least $2r+1$ and smoothness order $r$, the dimension of the spline space equals Schumaker's lower bound.

Optimality conjecture. The bound

Let $\mu\in C[-1,1]$ be a nonnegative majorant, and let $\omega_\mu\in\mathcal{P}_n$ be the snake polynomial associated with $\mu$, meaning that $|\omega_\mu(x)|\leq\mu(x)$ on…

Let $R$ be any ring. A module $F$ is flat when tensoring with $F$ preserves exact sequences, and a class of modules is covering when every module has a cover by an object of that c…

Input-adaptivity conjecture. The same linear improvement in approximation ability should hold even when the choice of inputs is allowed to depend on the function being approximated…

Let $\mathcal{P}_n$ be the real polynomials of degree at most $n$, let $\mu\geq 0$ be a majorant on $[-1,1]$, and define … Let $\omega_\mu$ denote the associated snake polynomial.…

Karlin's conjecture. For every $\delta>0$,

Let $\Delta$ be a triangulation of a simply connected polygonal domain in $\mathbb{R}^2$, and let $C^r_k(\Delta)$ be the vector space of splines of smoothness $r$ and degree at mos…

Let $f\in H(E)$, and let $\rho_n(f,E)$ be the error of best rational approximation on $E$, with asymptotic rate parameter $\rho(f,E)$. Gonchar's liminf conjecture. For every such f…

Let $X$ be a finite-dimensional Banach space, let $q$ be in the range assumed in the surrounding discussion, let $X^*$ have cotype constant $C_q(X^*)$, and let $w_2(X)$ be the seco…

Let $w_2(\ell_{\infty}^n)$ denote the second Whitney constant associated with the $n$-dimensional cube $Q^n$, and let $E_2(f;Q^n)$ and $\omega_2(f:Q^n)$ denote the corresponding qu…

Matrix-kernel approximation conjecture.

The squared Bernstein-factor conjecture. The factor $\alpha(K,{\bf x})$ in this estimate should be replaced by $\alpha(K,{\bf x})^2$, so that

The main conjecture. The Whitney constant satisfies $W_m=1$.

The exact Stechkin constant conjecture. For every $r\in\mathbb{N}$,

Fekete-array Monge–Ampère conjecture. The measures $\mu_n$ converge weak- to $\mu_K$.

Mergelyan property characterization. The domain $D$ has the Mergelyan property if and only if there are pseudoconvex domains $D_j$ with $\bar D\subset D_j$ such that

Erdélyi–Magnus–Nevai conjecture. The quantity $\mathcal{M}_k^{\alpha,\beta}$ satisfies

General lower-bound conjecture. The lower bound

Let ${\mathcal A}_{2m}^\mu$ be the reconstruction operator in Algorithm N/2, acting from $C(B^2)$ to $C(B^2)$, where $\mu$ is the attenuation parameter and $m$ tends to infinity. W…