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).


  • $\begingroup$ 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. $\endgroup$ – William DeMeo Feb 17 '12 at 4:31
  • $\begingroup$ @WilliamDeMeo Thanks for the precise reference. $\endgroup$ – 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.

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

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  • $\begingroup$ Where can I get it? Can you provide a reference? $\endgroup$ – becko Aug 2 '12 at 17:53
  • $\begingroup$ @becko: I included a link to prof. Dahmen's webpage. $\endgroup$ – timur Aug 3 '12 at 14:14

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