The wavelet transform is a tool that cuts up data or functions or operators into different frequency components, and then studies each component with a resolution matched to its scale. Forerunners of this technique were invented independently in pure mathematics (Calder n's resolution of the identity in harmonic analysis see e.g., Calder n (1964)), physics (coherent states for the ( ax + b)-group in quantum mechanics, first constructed by Aslaksen and Klauder (1968), and linked to the hydrogen atom Hamiltonian by Paul (1985)) and engineering (QMF filters by Esteban and Galland (1977), and later QMF filters with exact reconstruction property by Smith and Barnwell (1986), Vetterli (1986) in electrical engineering; wavelets were proposed for the analysis of seismic data by J. Morlet (1983)). The last five years have seen a synthesis between all these different approaches, which has been very fertile for all the fields concerned.

Let us stay for a moment within the signal analysis framework. (The discussion can easily be translated to other fields.) The wavelet transform of a signal evolving in time (e.g., the amplitude of the pressure on an eardrum, for acoustical applications) depends on two variables: scale (or frequency) and time; wavelets provide a tool for time-frequency localization. The first section tells us what time-frequency localization means and why it is of interest. The remaining sections describe different types of wavelets.