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I am looking for research that has been done in Discrete wavelets. Let me be specific as Google doesn't give me what I want when I say "discrete wavelets". I don't want countable basis for $ L^2(\mathbb{R}) $, Daubechies book, "Ten Lectures on Wavelets", already has this.

I am looking for research on $ \ell^2(\mathbb{R})$ vector spaces and wavelets that form either orthogonal or preferably non-orthogonal basis for them with a compact support in the frequency and space domains.

In the standard wavelet theory, the major benefit of the vectors is their compact space and frequency resolution. In addition to compact space support, I would like "over-completeness" (ie when $ A>1 $ for unit vectors, see below for $ A $) for the noise reduction property. Having a chain of vector spaces like in wavelet multi-resolution analysis would be nice as well, ie $ V_0 \subset V_1 \subset ... \subset \ell^2 $.

Or in math terms, I am looking for research on sets $W = \{ \psi_\alpha) \}_\alpha \subset \ell^2(\mathbb{C})$ with the following properties:

  1. $\operatorname{span}(W) = \ell^2(\mathbb{C})$;
  2. $\sum_\alpha \langle f, \psi_\alpha \rangle \langle \psi_\alpha, g \rangle = A \langle f, g \rangle$ for all $f, g \in \ell^2(\mathbb{C})$ and $A \geq 0$.

Recall $\ell^2(\mathbb{C}) = \left\{f\colon \mathbb{Z} \rightarrow \mathbb{C} \ \middle\vert \ \sum_{n\in \mathbb{Z}} |f(n)|^2 < \infty \right\}$ with inner product $\langle f, g \rangle = \sum_{n \in \mathbb{Z}} \overline{f(n)} g(n)$.

If it doesn't make sense, please let me know.

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    $\begingroup$ Check papers of Feichtinger, Grochenig, Balan, Casazza (among others) on Gabor multipliers for examples of Gabor systems in the discrete setting. Maybe later I'll try to look a few up. $\endgroup$ – mkreisel Nov 10 '15 at 17:28
  • $\begingroup$ Supposedly most people here know what $\ell^2$ is. I reorganized your question in a more organic way. Please pay more attention next time and do not put a bunch of centered formulas in one environment and remember LaTeX doesn't process text well (use \text{}). $\endgroup$ – Silvia Ghinassi Nov 10 '15 at 19:08
  • $\begingroup$ @WillieWong, I agree about the compact frequency support for $ \ell^2 $ vectors, but I would also like compact support in the space domain. In the standard wavelet theory, the major benefit of the vectors is their compact space and frequency resolution. In addition to compact space support, I would like "over-completeness" (ie when $ A > 1 $ for unit vectors) for the noise reduction property. Having a chain of vector spaces like in wavelet multi-resolution analysis would be nice as well, ie $ V_0 \subset V_1 \subset ... \subset \ell^2 $. $\endgroup$ – aidan.plenert.macdonald Nov 10 '15 at 20:35
  • $\begingroup$ Probably not at all what you want, but there are two papers by Benedetto and Benedetto (John and Rob) on $p$-adic wavelets. These wavelets seem rather discrete, due to the nature of the $p$-adic numbers. $\endgroup$ – Joe Silverman Nov 11 '15 at 1:45
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I followed the comment by mkreisel and discovered that I should have been Googling "Discrete-time Wavelet transform"

Here are links to a paper and a book,

  1. A Discrete-time Multiresolution Theory
  2. Wavelets and Sub-band Coding (see Chapter 3)

Thank you commenters

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