6
THE DAUBECHIES WAVELETS
The wavelets that we have looked at so far—Haar, Shannon, and linear spline wavelets—all have major drawbacks. Haar wavelets have compact support, but are discontinuous. Shannon wavelets are very smooth, but extend throughout the whole real line, and at infinity they decay very slowly. Linear spline wavelets are continuous, but the orthogonal scaling function and associated wavelet, like the Shannon wavelets, have infinite support; they do, however, decay rapidly at infinity.
These wavelets, together with a few others having similar properties, were the only ones available before Ingrid Daubechies discovered the hierarchy of wavelets that are named after her. The simplest of these is just the Haar wavelet, which is the only discontinuous one. The other wavelets in the hierarchy are both compactly supported and continuous. Better still, by going up the hierarchy, they become increasingly smooth; that is, they can have a prescribed number of continuous derivatives. The wavelet’s smoothness can be chosen to satisfy conditions for a particular application. We now turn to Daubechies’ construction of the first wavelet past Haar.
6.1 DAUBECHIES’ CONSTRUCTION
Theorem 5.23 lists the three sufficient conditions on the polynomial P that ensures that the iteration given in Section 5.3.4 produces a scaling function.
For a given polynomial P(z), let
In terms of the ...
