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.

我们证明了最佳非线性误差 ${\mathrm{大号}}_p$凸分区上的分段常数的近似值是 $\mathcal{Ø}（{ñ}^{-\frac{2}{d+\mathrm{1个}}}）$，在哪里 $ñ$ 是Sobolev空间中所有函数的像元数 ${\mathrm{w\; ^}}_q^2（\Omega ）$ 在一个立方体上 $\Omega \subset {\mathbb{[R}}^d$， $d⩾2$， 立刻 $\frac{2}{d+\mathrm{1个}}+\frac{\mathrm{1个}}{p}-\frac{\mathrm{1个}}{q}⩾0$。近似阶数$\mathcal{Ø}（{ñ}^{-\frac{2}{d+\mathrm{1个}}}）$在通过对自适应二元分区进行各向异性细化而获得的多面体分区上可以实现。对于各种Sobolev和Sobolev–Slobodeckij空间，可以从上下获得进一步的近似阶数估计${\mathrm{w\; ^}}_q^{\mathrm{[R}}（\Omega ）$ 嵌入 ${\mathrm{大号}}_p（\Omega ）$，其中一些还可以改善标准估算 $\mathcal{Ø}（{ñ}^{-\frac{\mathrm{1个}}{d}}）$ 已知在各向同性分区上是最佳的。