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:

- $\operatorname{span}(W) = \ell^2(\mathbb{C})$;
- $\sum_\alpha \langle f, \psi_\alpha \rangle \langle \psi_\alpha, g \rangle = A \langle f, g \rangle$ 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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). $\endgroup$ – Silvia Ghinassi Nov 10 '15 at 19:08