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Nonlinear piecewise polynomial approximation beyond Besov spaces. (English) Zbl 1032.41018
For $$0<p<\infty$$, $$n$$-term (nonlinear) approximation in $$L^p$$ and in two dimensions is studied, where the approximation is by piecewise polynomials and splines (e.g., Courant finite elements, but also discontinuous piecewise polynomials are used). The achieved rates of approximation are characterized by a certain family of smoothness spaces that are called B-spaces and that are generalizations of Besov spaces. The B-spaces stem from piecewise polynomials over triangulations that even allow arbitrarily sharp angles in the $$n$$ triangles which are used for the $$n$$-term approximation. Especially for approximations by functions with support on such triangles with sharp angles, the Besov spaces fail to be useful. Also, the triangulations used for the nonlinear approximation are nested subsets of the plane (multilevel triangulations). Both Bernstein and Jackson estimates are established. Also approximations that attain the best rates of approximation are generated explicitly.

##### MSC:
 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 41A15 Spline approximation 41A50 Best approximation, Chebyshev systems 41A05 Interpolation in approximation theory
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