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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).


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It is perhaps not very natural (I am not familiar with the subject), but maybe you could take a basis of wavelets for $\mathbb{L}^2 (0,1)$, and extend it to a basis for $\mathbb{L}^2 (0,\infty)$ by taking translations of each initial wavelet (that is, if $w_n$ is an element of the basis for $\mathbb{L}^2 (0,1)$, then each $w_n (\cdot-k)$, where $k$ is a natural number, would be an element of the basis for $\mathbb{L}^2 (0,\infty)$)? – D. Thomine Feb 15 '12 at 0:56
@D.Thomine: Yes. I think that's right. So the Haar system of wavelets can be used for $L^2[0,\infty)$. But I need continuous wavelets (see the edit I just made to the question). Thanks anyway (sorry that I didn't write the question right in the first place) – becko Feb 15 '12 at 23:27

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