**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 $\R^d$, $d>2$.

**Preprint version:** pdf

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