Journal of Complexity ( IF 1.338 ) Pub Date : 2019-11-06 , DOI: 10.1016/j.jco.2019.101444
Oleg Davydov; Oleksandr Kozynenko; Dmytro Skorokhodov

We prove that the error of the best nonlinear ${L}_{p}$-approximation by piecewise constants on convex partitions is $\mathcal{O}\left({N}^{-\frac{2}{d+1}}\right)$, where $N$ is the number of cells, for all functions in the Sobolev space ${W}_{q}^{2}\left(\Omega \right)$ on a cube $\Omega \subset {\mathbb{R}}^{d}$, $d⩾2$, as soon as $\frac{2}{d+1}+\frac{1}{p}-\frac{1}{q}⩾0$. The approximation order $\mathcal{O}\left({N}^{-\frac{2}{d+1}}\right)$ is achieved on a polyhedral partition obtained by anisotropic refinement of an adaptive dyadic partition. Further estimates of the approximation order from the above and below are given for various Sobolev and Sobolev–Slobodeckij spaces ${W}_{q}^{r}\left(\Omega \right)$ embedded in ${L}_{p}\left(\Omega \right)$, some of which also improve the standard estimate $\mathcal{O}\left({N}^{-\frac{1}{d}}\right)$ known to be optimal on isotropic partitions.

down
wechat
bug