Introduction

We have noted in previous chapters that the usual approach to the theory of wavelets makes use of the Fourier transform. Application of the transform allows all the problems we want to solve to be ‘translated’ into a different form, where a different range of techniques can usually be applied to help solve them; and since there is also a ‘translation’ back in the other direction, solutions obtained in this way give us solutions to the original problems.

Anything approaching a full discussion of this approach is impossible in this book. If the Fourier transform is to be discussed in any depth, a substantial body of other theory (Lebesgue integration theory, in particular) needs to be developed first. A proper discussion of the transform itself is then a non-trivial application of this background work, and the application to wavelets itself is again non-trivial. While all this work certainly repays the effort, this book is not the place to attempt it.

Nevertheless, the Fourier transform approach is the subject of this chapter. We aim to explore the structure of wavelet theory with the extra insight given by use of the Fourier transform, though necessarily without many proofs. As in the previous chapters, it is possible to proceed usefully in this way – as, for example, in our discussion of Lebesgue integration earlier, where we introduced just enough of the ideas and results to allow our discussion to proceed. Thus although the going will be somewhat easier for a reader with a deeper level of background knowledge than that relied on in earlier chapters, we do not assume such knowledge.

The Complex Case

Our attention so far in this text has been exclusively on real-valued structures: our vector and inner product space theory has been for spaces over the field of real numbers, and the specific space L2(∝) which has been the location for our wavelet theory is a space of real-valued functions. However, the Fourier transform is intrinsically an operator on complex-valued functions, so it is necessary for us to spend some time systematically shifting our focus from the real to the complex case.