2.4.1 Generalities on frames

As we saw in Chapter 1 for the 1-D case, the reproduction property (2.36) means that the information contained in the WT S( , a, ?) is highly redundant. As a consequence, we might hope that no content will be lost if we restrict the WT to a subset of the parameter space, in particular, a discrete subset (for instance, a lattice). Then the integral is replaced by a sum over a discrete (but infinite) family of wavelets :

Here too, and by the same reasoning, one is thus led to the introduction of frames. Whereas we have barely sketched this topic in Chapter 1, we will now give a fairly detailed treatment. Further information (albeit mostly in 1-D) may be found in [121, 122,Dau92].

Let us start with a precise definition. According to the terminology introduced by Duffin and Schaefer [156] in the context of nonharmonic Fourier series, one has:

Definition 2.4.1. A countable family of vectors { ? _n} in a Hilbert space is called a (discrete) frame if there are two positive constants A, B, with 0 < A ? B < ? , such that

The two constants A, B are the frame bounds . If A = B > 1 , the frame is said to be tight . If A = B = 1 , and ? _n = 1, ? n, the set {