Multivariate \(n\)-term rational and piecewise polynomial approximation
type: Item
instance of: scholarly article
title: Multivariate \(n\)-term rational and piecewise polynomial approximation
zbMATH Open document ID: 1019.41004
DOI: 10.1016/S0021-9045(02)00060-6
published in: Journal of Approximation Theory
publication date: 2003-05-25T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 1915649
zbMATH Keywords: nonlinear approximation
zbMATH Keywords: multivariate piecewise polynomial approximation
zbMATH Keywords: rational approximation
author: Pencho Petrushev
reviewed by: László Leindler
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1915649