Abstract
We prove that the error of the best nonlinear Lp-approximation by piecewise constants on convex partitions is O(N−2d+1), where N is the number of cells, for all functions in the Sobolev space Wq2(Ω) on a cube Ω⊂Rd, d⩾2, as soon as 2d+1+1p−1q⩾0. The approximation order O(N−2d+1) 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 Wqr(Ω) embedded in Lp(Ω), some of which also improve the standard estimate O(N−1d) known to be optimal on isotropic partitions.
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