Multivariate \(n\)-term rational and piecewise polynomial approximation
We study nonlinear approximation in \(L_p(\mathbb{R}^d)\) \((0<p< \infty, d>1)\) from (a) \(n\)-term rational functions, and (b) piecewise polynomials generated by different anisotropic dyadic partitions of \(\mathbb{R}^d\). To characterize the rates of each such piecewise polynomial approximation we introduce a family of smoothness spaces (B-spaces) which can be viewed as an anisotropic variation of Besov spaces. We use the B-spaces to prove Jackson and Bernstein estimates and then characterize the piecewise polynomial approximation by interpolation. Our main estimate relates \(n\)-term rational approximation with piecewise polynomial approximation in \(L_p(\mathbb{R}^d)\). This result enables us to obtain a direct estimate for \(n\)-term rational approximation in terms of a minimal B-norm (over all dyadic partitions). We also show that the Haar bases associated with anisotropic dyadic partitions of \(\mathbb{R}^d\) can be successfully utilized for nonlinear approximation. We give an effective algorithm for best Haar basis or best B-space selection.
multivariate piecewise polynomial approximation
scientific article; zbMATH DE number 1915649
scientific article