O. Davydov, Approximation by piecewise constants on convex partitions, J. Approx. Theory, 164 (2012), 346-352. doi:10.1016/j.jat.2011.11.001

Abstract: We show that the saturation order of piecewise constant approximation in $L_p$ norm on convex partitions with $N$ cells is $N^{-2/(d+1)}$, where $d$ is the number of variables. This order is achieved for any $f\in W^2_p(\Omega)$ on a partition obtained by a simple algorithm involving an anisotropic subdivision of a uniform partition. This improves considerably the approximation order $N^{-1/d}$ achievable on isotropic partitions. In addition we show that the saturation order of piecewise linear approximation on convex partitions is $N^{-2/d}$, the same as on isotropic partitions.

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