= 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 $L=\lceil2\alpha J\rceil$ 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>.
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