What systems of wavelets provide a frame for $L^2[0,\infty)$.
For example, the Haar system of wavelets provides a basis for $L^2[0,1]$, and the harmonic wavelets provide a basis for all of $L^2(R)$. I'm not sure if I can use any of these as a basis for $L^2(0,\infty)$
Edit: The Haar system can be used for $L^2[0,\infty)$ by successively translating it the right (see comment). But I need a wavelet frame for $L^2[0,\infty)$ that is both continuous and differentiable (I need second continuous derivatives). The Haar wavelets are discontinuous. Also it would be useful if the wavelet had compact support (in the time-domain or in the frequency-domain).