# Optimal approximation order of piecewise constants on convex partitions

## TL;DR

This paper establishes the optimal approximation order of piecewise constant functions on convex partitions for functions in Sobolev spaces, showing that anisotropic adaptive partitions achieve better rates than isotropic ones.

## Contribution

It proves the approximation error rate for nonlinear $L_p$-approximation by piecewise constants on convex partitions and introduces anisotropic refinement methods to attain optimal approximation orders.

## Key findings

- Error rate of $igO(N^{-rac{2}{d+1}}ig)$ for Sobolev functions.
- Anisotropic adaptive dyadic partitions achieve optimal approximation order.
- Improved estimates over standard isotropic partition results.

## Abstract

We prove that the error of the best nonlinear $L_p$-approximation by piecewise constants on convex partitions is $\mathcal{O}\big(N^{-\frac{2}{d+1}}\big)$, where $N$ the number of cells, for all functions in the Sobolev space $W^2_q(\Omega)$ on a cube $\Omega\subset\mathbb{R}^d$, $d\geqslant 2$, as soon as $\frac{2}{d+1} + \frac{1}{p} - \frac{1}{q}\geqslant 0$. The approximation order $\mathcal{O}\big(N^{-\frac{2}{d+1}}\big)$ 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 $W^r_q(\Omega)$ embedded in $L_p(\Omega)$, some of which also improve the standard estimate $\mathcal{O}\big(N^{-\frac 1d}\big)$ known to be optimal on isotropic partitions.