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Approximation by piecewise constants on convex partitions

Davydov, O. (2012) Approximation by piecewise constants on convex partitions. Journal of Approximation Theory, 164 (2). pp. 346-352. ISSN 0021-9045

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We show that the saturation order of piecewise constant approximation in Lp 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 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.