By D. Ter Harr

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Algorithms and Error Bounds for Multivariate Piecewise Constant Approximation Oleg Davydov Department of Mathematics and Statistics, University of Strathclyde, G1 1XH, UK Summary. We review the surprisingly rich theory of approximation of functions of many variables by piecewise constants. This covers for example the Sobolev-Poincar´e 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.

We further remark that a natural image can always be represented by a bounded function. However, the converse is (trivially) not true: for any ﬁxed p ∈ [1, ∞], functions in Lp (Ω) do often not correspond to natural images. One of the main tasks of functional analysis methods in image processing is to deﬁne function classes, being given by some suitable regularity conditions, which are as small as possible but contain relevant images. In the context of triangulation methods, this immediately leads us to one central question: which image classes may be well-recovered by approximation methods relying on triangular meshes?

16. B. P. Almond, J. Luo, and R. Stevens: An evaluation of random RC networks for modelling the bulk ac electrical response of ionic conductors. Solid State Ionics 126(1), 1999, 65–80. Algorithms and Error Bounds for Multivariate Piecewise Constant Approximation Oleg Davydov Department of Mathematics and Statistics, University of Strathclyde, G1 1XH, UK Summary. We review the surprisingly rich theory of approximation of functions of many variables by piecewise constants. This covers for example the Sobolev-Poincar´e 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.

### Elements of Hamiltonian Mechanics by D. Ter Harr

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