I am looking for an orthogonal wavelet basis for a Hilbert Space $H$ of complex-valued square-integrable functions defined over the non-negative numbers. The inner product on $H$ is the weighted inner product:
$$\langle f; g \rangle_{w} = \int_0^{+\infty} f(x) \,\overline{g(x)}\, w(x) \, \mathrm{d}x$$
where $w$ is a positive function of $x$, $x \geq 0$, and $f, g \in H$. Additionally, I have the constraint that my wavelets are twice differentiable. Could someone suggest such a wavelet basis?