Chapter 8: Symmetry for Compactly Supported Wavelet Bases

All the examples we have seen so far of compactly supported orthonormal wavelet bases are conspicuously non-symmetric, in contrast to the infinitely supported wavelet bases we saw before, such as the Meyer and Battle-Lemari bases. In this chapter we discuss why this asymmetry occurs, what can be done about it, and whether anything should be done about it.

8.1. Absence of symmetry for compactly supported orthonormal wavelets.

In Chapter 5 we already saw that a multiresolution analysis does not determine ?, ? uniquely. This is again borne out by the following lemma.

Lemma 8.1.1.

If f _n( x) = f( x ? n) and , constitute orthonormal bases of the same subspace E of , then there exists a 2 ?-periodic function ?( ?), with ?( ?) = ? g ? ² = 1, so that .

Proof.

Since the f _n are an orthonormal basis for E ? g, g = ? _n ? _n f _n, with ? _n ? _n ² = 1. Consequently, , with ?( ?) = ? _n ? _n e ^{? in ?}.
As shown in Chapter 5, orthonormality of the f( ? n) is equivalent with a.e. Similarly . It follows that ?( ?) = 1.