Journal article
Authors list: Davydov, O; Petrushev, P
Publication year: 2003
Pages: 708-758
Journal: SIAM Journal on Mathematical Analysis
Volume number: 35
Issue number: 3
ISSN: 0036-1410
eISSN: 1095-7154
Open access status: Green
DOI Link: https://doi.org/10.1137/S0036141002409374
Publisher: Society for Industrial and Applied Mathematics
Abstract:
We study nonlinear n-term approximation in L-p(R-2) (0 < p <=infinity) from hierarchical sequences of stable local bases consisting of differentiable (i.e., C-r with r >= 1) piecewise polynomials (splines). We construct such sequences of bases over multilevel nested triangulations of R-2, which allow arbitrarily sharp angles. To quantize nonlinear n-term spline approximation, we introduce and explore a collection of smoothness spaces (B-spaces). We utilize the B-spaces to prove companion Jackson and Bernstein estimates and then characterize the rates of approximation by interpolation. Even when applied on uniform triangulations with well-known families of basis functions such as box splines, these results give a more complete characterization of the approximation rates than the existing ones involving Besov spaces. Our results can easily be extended to properly defined multilevel triangulations in R-d, d > 2.
Citation Styles
Harvard Citation style: Davydov, O. and Petrushev, P. (2003) Nonlinear approximation from differentiable piecewise polynomials, SIAM Journal on Mathematical Analysis, 35(3), pp. 708-758. https://doi.org/10.1137/S0036141002409374
APA Citation style: Davydov, O., & Petrushev, P. (2003). Nonlinear approximation from differentiable piecewise polynomials. SIAM Journal on Mathematical Analysis. 35(3), 708-758. https://doi.org/10.1137/S0036141002409374
Keywords
Jackson and Bernstein estimates; LINEAR INDEPENDENCE; multilevel bases; multilevel nested triangulations; multivariate splines; Nonlinear approximation; PREWAVELETS; SPLINES; STABLE LOCAL BASES; stable local spline bases
