Question

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.

Answer 1

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.

Reference: Wolfgang Dahmen, Angela Kunoth, Karsten Urban, Biorthogonal Spline-Wavelets on the Interval - Stability and Moment Conditions

It can be downloaded from http://www.igpm.rwth-aachen.de/node/223