Optimal stable manifold widths of Besov balls
Let $d \ge 1$, $1 \le p, q, r \le \infty$ and $s > 0$ with $s > d(1/q - 1/p)$, where $1/\infty = 0$. Let $\Omega = [0,1]^d$ and let $K$ be the unit ball of the Besov space $B^s_{q,r}(\Omega)$ with the norm $$\|f\|_{B^s_{q,r}} = \|f\|_{L^q(\Omega)} + \left( \int_0^1 \bigl[ t^{-s} \omega_m(f,t)_q \bigr]^r \, \frac{dt}{t} \right)^{1/r},$$ with the second term replaced by $\sup_{0 < t \le 1} t^{-s} \omega_m(f,t)_q$ when $r = \infty$. Here $m = \lfloor s \rfloor + 1$ and $\omega_m(f,t)_q = \sup_{|h| \le t} \|\Delta_h^m f\|_{L^q(\Omega_{mh})}$ is the modulus of smoothness of order $m$, with $\Delta_h^m$ the $m$th forward difference with step $h$ and $\Omega_{mh} = \{x \in \Omega : x + mh \in \Omega\}$. Then $K$ is a compact subset of $L^p(\Omega)$. For $\gamma \ge 1$ and $n \ge 1$ the stable manifold width of $K$ is $$\delta^*_{n,\gamma}(K)_{L^p} = \inf_{\|\cdot\|_Y, a, M} \, \sup_{f \in K} \|f - M(a(f))\|_{L^p(\Omega)},$$ the infimum taken over norms $\|\cdot\|_Y$ on $\mathbb{R}^n$ and maps $a : K \to (\mathbb{R}^n, \|\cdot\|_Y)$ and $M : (\mathbb{R}^n, \|\cdot\|_Y) \to L^p(\Omega)$ that are $\gamma$-Lipschitz on their whole domains; the norm and the maps may depend on $n$. Cohen, DeVore, Petrova and Wojtaszczyk introduced these widths and proved $\delta^*_{n,\gamma}(K)_{L^p} \ge c\, n^{-s/d}$ for such balls, from the decay of their entropy numbers; for $p = 2$ their bound of stable manifold widths in a Hilbert space by entropy numbers gives the matching upper bound, with $\gamma = 2$. For the manifold widths of DeVore, Howard and Micchelli, in which $a$ and $M$ need only be continuous, DeVore, Kyriazis, Leviatan and Tikhomirov proved the upper bound $C n^{-s/d}$, and Siegel the matching lower bound, for every such ball.
Problem. For every such $(d, s, p, q, r)$, are there constants $\gamma < \infty$ and $C < \infty$, independent of $n$, such that $$\delta^*_{n,\gamma}(K)_{L^p} \le C n^{-s/d} \qquad (n \ge 1)?$$
Source: Cohen, DeVore, Petrova and Wojtaszczyk (2022), §6.4, for the Besov and Sobolev balls compactly embedded in L^p; stated for Besov balls on the unit cube by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- FA
- Keywords
- AIM Open Applied Problems, Besov spaces, nonlinear approximation, nonlinear widths
References
@article{cohen2022,
author = {Cohen, Albert and DeVore, Ronald and Petrova, Guergana and Wojtaszczyk, Przemys{\l}aw},
title = {Optimal stable nonlinear approximation},
journal = {Found. Comput. Math.},
volume = {22},
number = {3},
year = {2022},
pages = {607--648},
doi = {10.1007/s10208-021-09494-z},
eprint = {2009.09907},
archivePrefix = {arXiv},
}
@article{devore1989,
author = {DeVore, Ronald A. and Howard, Ralph and Micchelli, Charles},
title = {Optimal nonlinear approximation},
journal = {Manuscripta Math.},
volume = {63},
number = {4},
year = {1989},
pages = {469--478},
doi = {10.1007/BF01171759},
}
@article{devore1993,
author = {DeVore, Ronald A. and Kyriazis, George and Leviatan, Dany and Tikhomirov, Vladimir M.},
title = {Wavelet compression and nonlinear $n$-widths},
journal = {Adv. Comput. Math.},
volume = {1},
number = {2},
year = {1993},
pages = {197--214},
doi = {10.1007/BF02071385},
}
@article{siegel2024,
author = {Siegel, Jonathan W.},
title = {Sharp lower bounds on the manifold widths of {S}obolev and {B}esov spaces},
journal = {J. Complexity},
volume = {85},
year = {2024},
pages = {101884},
doi = {10.1016/j.jco.2024.101884},
eprint = {2402.04407},
archivePrefix = {arXiv},
}
@misc{aim-stable-manifold-widths-besov,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Optimal stable manifold widths of Besov balls}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/542-stable-manifold-widths-besov.md},
note = {Entry 542 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00388, "Optimal stable manifold widths of Besov balls", clopen.solutions.