10 problems

Let $d\geq2$, let $r\in\mathbb N$, let $\lambda$ be an even integer with $\lambda>4$, and let $\varrho_n(\boldsymbol W^r_2(\mathbb R^d;\mu),L_2(\mathbb R^d;\mu))$ denote the sampli…

Let $d$ be the dimension, let $r\in\mathbb N$, let $\lambda$ be an even integer with $\lambda>4$, and let $W^r_2(\mathbb R^d;\mu)$ and $\mathcal H^{r_\lambda}(\mathbb R^d)$ denote…

Let $d$ be the dimension, let $r\in\mathbb N$, let $\rho$ be an even integer with $\rho>4$, and let $W^r_2(\mathbb R^d;\mu)$ and $\mathcal H^{r_\rho}(\mathbb R^d)$ denote the weigh…

Let $B^{\boldsymbol{r}}_{\infty,\theta}$ be the Nikol'skii–Besov class of periodic functions with mixed smoothness, let $L_\infty$ be the supremum-norm space, let $B_{\infty,1}$ be…

Let $B^{\boldsymbol{r}}_{\infty,\theta}$ be the Nikol'skii–Besov class of periodic functions with mixed smoothness, let $Q_n^{\boldsymbol{\gamma}}$ denote the relevant hyperbolic-c…

The conjecture. The same right asymptotic order should hold for $r=1$:

Let $d\in\mathbb{N}$, let $\mathbb{T}^d$ be the $d$-dimensional torus, and let $\boldsymbol{W}^r_p(\mathbb{T}^d)$ and $\boldsymbol{H}^r_p(\mathbb{T}^d)$ denote the mixed Sobolev an…

Let $\mathbf W^r_{q,\alpha}$ denote the mixed smoothness function class on the $d$-dimensional domain, and let $\varepsilon_k(\mathbf W^r_{q,\alpha},L_\infty)$ be its $k$th entropy…

Cubature error conjecture. For all $\alpha>1/p$,

Let the two spaces in the inclusion stated in Lemma $\mathrm{fullEmbed}$ be the full-domain function spaces defined earlier in the paper. Reverse-inclusion conjecture. The reverse…