Wavelets: Theory and Applications

1

The Continuous Wavelet Transform

In this chapter the wavelet transform is introduced and interpreted as a linear map between weighted L²-spaces. Its isometry leads immediately to an inversion formula based on the adjoint operator. The explicit calculation of this formula permits the use of different wavelets for the analysis and synthesis of signals. This coincides with the use of biorthogonal systems with respect to the discrete wavelet transform. Then we present some results which allow different interpretations of the wavelet transform.

First of all invariants are calculated. Using the filter properties we progress to the concept of phase-space representation and localization operators, which have many areas of use, particularly in physics. The approximation properties play an important role in the classification of different wavelets.

After these fundamental precepts the wavelet transform is considered in a more general setting. Group-theoretical considerations, which let us generalize the continuous wavelet transform in higher dimensions, end the first chapter.

1.1   Definition and Elementary Properties

The aim of this chapter is to extract specific information from a given function ƒ, called the signal. For this there is essentially one method: the signal is transformed in a suitable way in the hope that the desired information will then be easier to read. The transformation used naturally depends on the nature of the information we are interested in. Moreover, we want ...