TABLE OF CONTENTS This chapter is devoted to a brief examination of two topics. The first concerns a certain minimality property of gaborettes and how it generalizes to wavelets in one and two dimensions. The second is an analysis of the Wigner transform, as an alternative to the wavelet transform. This latter transform is extensively used in certain physical computations and in the analysis of radar signals. Notice that neither of these topics is a prerequisite for the study of the more general wavelets described in Chapters 9 and 10.
The generalized gaborettes defined in (6.123), which give rise to holomorphic Gabor transforms, have a well-known minimal uncertainty property, related to localization in phase space. In (7.23) we had introduced the localization operators a ?( ?). As discussed in Chapter 7, these operators can be used to measure the proportion of the signal transform S which is concentrated in the (phase space) region ?. Consider the case of Gabor wavelets and let ? q,p ? L 2( 
, dx) be the family of gaborettes defined in (6.109), using the window function ?. These vectors satisfy the resolution of the identity (6.112). Assuming the normalization ? 2 = 1 /2 ?, we see that the operators
give rise to the probability measure
for any signal s ? L 2( , dx) with Gabor transform S. In Section 6.3 it was...
