Nonlinear Approximation from Differentiable Piecewise Polynomials

We study nonlinear n-term approximation in $L_p({\mathbb R}^2)$ ($0<p\le\infty$) from hierarchical sequences of stable local bases consisting of differentiable (i.e., $C^r$ with $r\ge1$) 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.

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