Best N-term wavelet approximation of piecewise Hölder functions

For a bounded function on an interval with finitely many jumps and piecewise $C^{\alpha }$ bounds, localized orthonormal wavelets with sufficient vanishing moments give squared best N-term approximation error $O(N^{-2\alpha })$. There are $O(2^j)$ smooth-region coefficients of size $O(2^{-j(\alpha +1/2)})$, and only $O(K)$ jump-crossing coefficients of size $O(2^{-j/2})$ per level. Retain all coefficients through level $J$ and the jump-crossing coefficients through $L=\lceil 2\alpha J\rceil$ when $\alpha >1/2$. This costs $O(2^J+KJ)$ and leaves squared tails $O(2^{-2\alpha J}+2^{-L})$. Choosing $2^J$ comparable to $N$ proves the rate by the Parseval identity.