Abstract: We study nonlinear $n$-term approximation in $L_p(\R^2)$ ($0 < p \le \infty$) from hierarchical sequences of stable local bases consisting of differentiable (ie $C^r$ with $r \ge 1$) 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 $\R^d$, $d>2$.
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