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MULTIRESOLUTION ANALYSIS

A multiresolution analysis (MRA) for a space L is tower of subspaces

together with a scaling function (or "father wavelet") such that
The above definition can be generalized in several ways. We can replace condition 2 by an apparently more general condition:
2*.
(Riesz basis) {(x-j):j} is a Riesz basis.
But in fact, it can be shown that in the presense of the the other conditions, condition 2 and 2* are interchangeable. This gives us some flexibility in choosing for an MRA. See [3] (p. 21) for some basic facts about Riesz basis, and more generally the concept of a frame.

We can also work with functions f(x) over d (d=1,2,...). Then (following [Strichartz]) we can replace by a lattice d. Then the scaling property can be replaced by

where M is a linear transformation of d satisfying M and all the eigenvalues of M satisfy ||>1. In this case |detM| is an integer m2 and /M is a group of order m. In the simplest case, =d is the unit lattice and M=2I with m=2d.

These conditions for MRA are also not independent: thus, [2] shows that condition 3 (separation) is implied by conditions 1 and 2*. Assuming that the Fourier transform of has the property that |F()(y)| is continuous at y=0. Then

NOTATION: in the literature, there are two conventions for indexing the tower of subspaces in an MRA. Mallat uses the convention ViVi-1 while our notation follows Daubechies. Let us write V-i for Vi so that we have ViVi-1, to get an indexing consistent with Mallat. The naturalness of each indexing scheme (Vi or Vi) can be explained this way. If you are more interested in the analysis of signals, then the Mallat convention makes more sense. That is, you begin with a signal f=f0V0, and you can produce successively coarser approximations f1V1, f2V2, etc. Following Mallat, we call the superscript i in Vi the "scale" of fi, and this is the inverse of the "resolution". If you want to do signal synthesis, then you begin with a signal f=f0V0 and by adding more details, you can refine it to f1V1, f2V2, etc.


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