In Ten Lectures on Wavelets Chapter 3.3, we have the following claim:
if $\psi_{m,n}$ constitute a tight frame and also form an orthogonal basis of $L^2(\mathbb{R})$ , we have equation (3.3.8) $$\int_0^\infty \xi^{-1}|\hat{\psi}(\xi)|^2 d\xi = \frac{b_0\ln a_0}{2\pi}$$ for some constant $a_0, b_0$, and $\psi$ is the corresponding mother wavelet.
Haar wavelet, in particular, satisfies this conditions (orthonormal basis), and should satisfy the equation above, according to Daubechies remark right after this equation. Note that Haar function has the following form: $$\psi(t) = \begin{cases} 1, \; t \in [0, \frac{1}{2}) \\ -1, t \in [\frac{1}{2}, 1) \\ 0, \; \text{otherwise}\end{cases}$$
But I am having trouble showing it. In fact, I have obtained that $\xi^{-1}|\hat{\psi}(\xi)|^2 = \frac{1}{\xi^3\pi}(3 - 4\cos\frac{\xi}{2} + 2\cos\xi)$, which seems to diverge in the integral above. Is there anything wrong with my computation or my perception of this concept? Thank you!