# Is there a wavelet frame for $L^2[0,\infty)$?

What systems of wavelets provide a discrete frame for $L^2[0,\infty)$?

Specifically, I need a mother wavelet $\psi(x)$ that has a continuous second derivative, such that the system of wavelets $\{\psi_{jk}(x)=\psi(2^jx-k)|j,k\in\mathbb{Z}\}, k\geq0$ forms a basis (or generally a frame) for $L^2[0,\infty)$. Preferably $\psi(x)$ should have compact support, and the wavelets $\psi_{jk}(x)$ should be zero for $x<0$.

If the proposed wavelets $\psi_{jk}(x)$ are not orthogonal, then I'll also need the reciprocal basis (or frame).

Thanks.

• Take your favorite wavelet basis for $L^2(\mathbb R)$. You can construct a basis for $L^2[0,\infty)$ using the procedure described in Daubechies, "Ten Lectures on Wavelets," pages 334-335. – William DeMeo Feb 17 '12 at 4:31
• @WilliamDeMeo Thanks for the precise reference. – becko Feb 18 '12 at 5:50

There is a general construction by Dahmen-Kunoth-Urban that adapts Cohen-Daubechies-Feauveau wavelets to $[0,\infty)$ or $[0,1]$ satisfying various boundary conditions.