Wavelets: Theory and Applications

Preface

Wavelets have undergone a rapid growth in the last fifteen years both in research and applications. As often happens, wavelets were first developed to solve an engineering problem that could not be satisfactorily solved with traditional techniques. The failure of classical methods for analysing geophysical data was the starting point for the development of a ‘new’ analysing tool: wavelet analysis. Again, as in many other cases, time has shown that this tool is based on a powerful mathematical theory. The interplay of application and mathematical analysis is the basis of the wavelet success story.

The main disadvantage of the classical tool in signal processing, namely the Fourier transform, is its missing localization property: if a signal changes at a specific time, its transform changes everywhere and a simple inspection of the transformed signal does not reveal the position of the alteration. The reason is, of course, the periodic behaviour of the trigonometric functions. If we use, instead, as analysing function a locally confined little wave, a wavelet (or ondelette as they are called in France, where they originated), then translation and scaling allows for a frequency resolution at arbitrary positions.

Both continuous and discrete wavelet transforms were developed at about the same time.

The continuous wavelet transform can be viewed as a phase-space representation. Filter and approximation properties are studied. The group representation approach allows for rather ...