Hölder-Taylor remainder bound ($|f(x)-T_{x_0}(x)|\le C\Vert f{\Vert }_{C^{\alpha }}|x-x_0{|}^{\alpha }$)

Write $\alpha =r+\beta$ with an integer $r\ge 0$ and $0<\beta \le 1$, using $r=\alpha -1$ at integer $\alpha$. For $f\in C^{r,\beta }$, subtract its degree-$r$ Taylor polynomial at $x_0$. For $r\ge 1$, the integral Taylor remainder with $f^{(r)}(t)-f^{(r)}(x_0)$ is bounded using the Hölder seminorm; for $r=0$ this is exactly Hölder continuity. The resulting bound is the displayed estimate. At integer order, the convention is $C^{r,1}$; a $C^{r+1}$ function on a compact closed interval also obeys this estimate.

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.