Nonlinear Approximation from Differentiable Piecewise Polynomials

We study nonlinear n-term approximation in Lp(R) (0 < p ≤ ∞) from hierarchical sequences of stable local bases consisting of differentiable (i.e., Cr with r ≥ 1) piecewise polynomials (splines). We construct such sequences of bases over multilevel nested triangulations of R2, which allow arbitrarily sharp angles. To quantize nonlinear nterm 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 Rd, d > 2.