Wavelet Transforms And Their Applications To Turbulence

Wavelet transforms are recent mathematical techniques, based on group theory and square integrable representations, which allows one to unfold a signal, or a field, into both space and scale, and possibly directions. They use analyzing functions, called wavelets, which are localized in space. The scale decomposition is obtained by dilating or contracting the chosen analyzing wavelet before convolving it with the signal. The limited spatial support of wavelets is important because then the behavior of the signal at infinity does not play any role. Therefore the wavelet analysis or synthesis can be performed locally on the signal, as opposed to the Fourier transform which is inherently nonlocal due to the space-filling nature of the trigonometric functions. Wavelet transforms have been applied mostly to signal processing, image coding, and numerical analysis, and they are still evolving. So far there are only two complete presentations of this topic, both written in French, one for engineers (Gasquet & Witomski 1990) and the other for mathematicians (Meyer 1990a), and two conference proceedings, the first in English (Combes et al 1989), the second in French (Lemari6 1990a). In preparation are a textbook (Holschneider 1991), a course (Daubechies 1991), three conference proceedings (Meyer & Paul 1991, Beylkin et al 1991b, Farge et al 1991), and a special issue of IEEE Transactions