Ten Lectures on Wavelets
This text contains 10 lectures given by the author at the CBMS conference on wavelets organized in June 1990 by the Mathematics Department at the University of Lowell, Massachusetts.
TABLE OF CONTENTS Ten Lectures on Wavelets
Chapter 10: Generalizations and Tricks for Orthonormal Wavelet Bases
This chapter consists of several generalizations and extensions of earlier constructions. These are not treated with the same detail as in the previous chapters. Some of the topics are still developing, and I expect that any write-up on them would look very different even two years from now. The sections cover multi-dimensional wavelets with dilation factor 2, via tensor product multiresolution analysis, or via nonseparable schemes; orthonormal wavelet bases with dilation factor different from 2, integer or non-integer; the "splitting trick" for better frequency resolution (in fact, merely a special case of the "wavelet packets" of Coifman and Meyer); wavelet bases on an interval.
10.1. Multidimensional wavelet bases with dilation factor 2.
For simplicity, we will consider only the two-dimensional case; higher dimensions are analogous. One trivial way of constructing an orthonormal basis for 
, starting from an orthonormal wavelet basis ? j,k( x) = 2 ? j/2 ?(2 j x ? k) for 
, is simply to take the tensor product functions generated by two one-dimensional bases:
The resulting functions are indeed wavelets, and 
is an orthonormal basis for 
. In this basis the two variables x 1 and x 2 are dilated separately.
There exists another construction, more interesting for many applications, in which the dilations of the resulting orthonormal wavelet basis control both variables simultaneously. In this construction, one considers the tensor product of two one-dimensional multiresolution analyses rather than of the corresponding wavelet bases. More precisely, define...
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