O. Davydov, Algorithms and error bounds for multivariate piecewise constant approximation, in "Approximation Algorithms for Complex Systems," (E. H. Georgoulis, A. Iske and J. Levesley, Eds.), Springer Proceedings in Mathematics, Vol. 3, pp. 27--45, Springer-Verlag, 2011. doi:10.1007/978-3-642-16876-5_2

Abstract: We review the surprisingly rich theory of approximation of functions of many variables by piecewise constants. This covers for example the Sobolev-Poincaré inequalities, parts of the theory of nonlinear approximation, Haar wavelets and tree approximation, as well as recent results about approximation orders achievable on anisotropic partitions.

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