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Nonlinear approximation from differentiable piecewise polynomials

Davydov, O. and Petrushev, P. (2003) Nonlinear approximation from differentiable piecewise polynomials. SIAM Journal on Mathematical Analysis, 35 (3). pp. 708-758. ISSN 0036-1410

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Abstract

We study nonlinear $n$-term approximation in $L_p({mathbb R}^2)$ ($0<pleinfty$) from hierarchical sequences of stable local bases consisting of differentiable (i.e., $C^r$ with $rge1$) piecewise polynomials (splines). We construct such sequences of bases over multilevel nested triangulations of ${mathbb R}^2$, 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 ${mathbb R}^d$, d>2.