Wavelet coefficient decay for Hölder functions

Localized wavelets of support diameter $O(2^{-j})$ and $L^1$ norm $O(2^{-j/2})$, with enough vanishing moments, satisfySubtract a Taylor polynomial annihilated by the wavelet, apply the Hölder-Taylor remainder bound on its support of a function, and multiply by its $L^1$ norm. The same argument applies to localized boundary wavelets with the same cancellation; finitely many coarse scaling functions are handled separately.