46 problems

Let $D\subset\mathbb{R}^d$ be a convex compact set. A sequence of finite subsets $\{Y_n\}_{n\ge1}$ of $D$ is an optimal polynomial mesh if there are constants $\mu,\nu>0$, dependin…

Let $E\subset\mathbb R$ be a compact uniformly perfect set, and let $C(E)$ denote the continuous functions on $E$. The source's Problem 8 asks whether membership in a modulus-of-co…

Let ${\mathcal F}_n \subset {\mathcal P}_n({\mathbb Z})$ be the set of irreducible polynomials over ${\mathbb Z}$ of exact degree $n$ whose zeros all lie in $[0,1]$. For…

Let $H^p$ denote the Hardy space on the unit disk, let $d54b$ be the unit circle, and let $f\in H^p$ satisfy $f(0)\neq 0$. An inner function is a bounded analytic function on the u…

Let $K$ be a compact subset of the complex plane, and write $K^\circ$ for its interior. A continuous function $f:K\to\mathbb C$ is zero-free on $K^\circ$ when $f(z)\neq 0$ for ever…

Let $K\subset {\mathbb{C}}\setminus \{0\}$ be a compact set. A polynomial of the indicated form is $\sum_{k=n/\tau_n}^{n}a_kz^k$, where $\tau_n\to 1$. Polar-set characterization co…

Let $K$ be a non-empty compact subset of the complex plane, and let $\delta_n$ and $C$ denote the quantities defined in the preceding results. Dilation asymptotic conjecture. … Thi…

Let $\Pi_d$ be the polynomial set associated with degree-$d$ approximation conditions, and let $\overline{\Pi^*_{2^7}}$ denote the closure of the polynomial set obtainable with sev…

Let $\Pi_d$ be the polynomial set associated with degree-$d$ approximation conditions, and let $\overline{\Pi^*_{2^6}}$ denote the closure of the polynomial set obtainable with six…

Let $\Pi_d$ be the polynomial set associated with degree-$d$ approximation conditions, and let $\overline{\Pi^*_{2^5}}$ denote the closure of the polynomial set obtainable with fiv…

Let $\mu_1$ be the double-sided exponential measure on $\mathbb{R}$ with density $e^{-\lvert x\rvert}/2$. For an absolutely continuous function $f$, let $f_k$ denote its polynomial…

Let $\boldsymbol{\Omega}$ be the compact support of a positive Borel probability measure $\mu$ on $\mathbb{R}^d$, and let $f$ be its positive, continuous density with respect to Le…

Let $O_{N,5}$ be the optimized model with approximation order $N$ and entropy-approximation parameter $5$, optimized on the interval $[-5,5]$. Let $O_5(x)$ denote the corresponding…

Let $O_{2K+1}$ denote the optimized polynomial approximation model of degree $2K+1$, and let $\beta_{BS}$ denote the Boltzmann–Shannon entropy function. Fix an interval of the real…

Let $\mu$ be a measure of the form described in the paper, with density $w$ on the relevant part of the unit circle and associated function $G$. For every interval $I$, suppose tha…

Asymptotic expansion conjecture. As $k\to\infty$,

Let $1<p<\infty$, $f\in H^p$, and $n\in\mathbb{N}$. If $f(0)\neq 0$, then the optimal polynomial approximant $q_{n,p}[f,1]$ is the polynomial of degree at most $n$ that best approx…

Let $E_n(\alpha)$ denote the minimax error for degree-$n$ polynomial approximation to the checkmark function, and let its V-shapes be the intervals described in the preceding conje…

Let $E_n(\alpha)$ be the minimax error for degree-$n$ polynomial approximation to the checkmark function, with parameter $\alpha\in(-1,1)$. Shekhtman's conjecture. For odd $n$,…

Let $E_n(\alpha)$ denote the minimax error for degree-$n$ polynomial approximation to the checkmark function, as a function of the parameter $\alpha$. A V-shape is a contiguous int…

Square-root-degree conjecture. Using arguments of Newman and Rivlin, $N=O(\sqrt{q})$ should be sufficient to accurately approximate $\mathbf{A}^p$.

Let $K$ be a compact subset of $\mathbb C$ with connected complement, and let $g$ be a continuous function on $K$, holomorphic in $K^0$, with no isolated zeros in $K^0$. Zero-free…

Let $K$ be a compact set with connected complement, and let $f$ be a continuous function on $K$ that is analytic in the interior $K^\circ$ and zero-free on $K^\circ$. Lavrentiev–An…

Let $f(x)\in\mathbb{Z}[x]$ have degree $n\geq 0$. A polynomial is squarefree if it is not divisible by the square of an irreducible polynomial over $\mathbb{Q}$. Define…

Let $f(x)\in\mathbb{Z}[x]$ have degree $n\geq 1$. Define $L(f-g)=\sum_{j=0}^{n}|b_j-a_j|$ when $f(x)=\sum_{j=0}^{n}a_jx^j$ and $g(x)=\sum_{j=0}^{n}b_jx^j$. Turán's distance-two con…