= Wavelet approximation across a curve
A finite-length smooth curve meets $O(2^j)$ supports of localized two-dimensional <wavelets> at scale $j$. For a bounded function, their coefficients are $O(2^{-j})$, so their total squared energy at that scale is $O(2^{-j})$. Keeping these coefficients through level $J$ uses $O(2^J)$ terms and leaves squared error $O(2^{-J})$. Polynomial cancellation, or adequate approximation of the remaining smooth regions, therefore gives <best N-term approximation> squared error $O(N^{-1})$.
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