Wavelets, Fractals and Fourier Transforms

Recently there have been many developments and new applications of mathematical techniques for describing complex algebraic functions and analyzing empirical continuous data derived from many different types of signal, for example turbulent flows, oil well logs and electrical signals from the eye. Probably the most important and rapidly developing of these techniques involve Fourier methods (the oldest - nearly 200 years), fractals (about 30 years old in practice, though 80 in theory) and wavelets (about 15 years old). An international conference on these developments was organized jointly by the Societe de Mathemetiques et Industrielles (SMAI) and co-sponsered by the European Research Council of Fluid Turbulence and Combustion (ERCOFTAC), Trinity College and the US Air Force. It was held at Newnham College, Cambridge, in December 1990. Readers of this volume should find that these papers provide a useful introduction to the mathematics of wavelets, fractals and Fourier transforms, and to their many applications. They should realize that the different methods of analysis expose different aspects of complex signals and surfaces and that the most suitable method often depends on the application under construction.

Table of Contents:
Part 1: wavelets, fractals and Fourier transforms - detection and analysis of structure, J.C.R. Hunt et al; wavelets, fractals and order-two densities, K.J. Falconer; orthonormal and continuous wavelet transform - algorithms and applications to the study of pointwise properties of functions, S. Jaffard; iterated function systems and their applications, J. Stark and P. Bressloff; biothogonal bases of symmetric compactly supported wavelets, C. Herley and M. Vetterli; fractional Brownian motion and wavelets, P. Flandrin; the wavelet Gibbs phenomenon, H.O. Raasmussen; multiscale segmentation of well logs, P.L. Vermeer and J.A.H. Alkemade. Part 2: scale-invariance and self-similar "wavelet" transforms - an analysis of natural scenes and mammalian visual systems, D.J. Field; wavlets and astronomical image analysis, A. Bijaoui and A. Fresnel; universe heterogeneities from a wavelet analysis, A. Bijaoui et al; the wavelet transformation applied to flow around Antarctica, B. Sinha and K.J. Richards; quantification of scale cascades in the stratosphere using wavelet transforms, P.H. Haynes and W.A. Norton. Part 3: multiple-scale correlation detection, wavelet transforms and multifractal turbulance, J.G. Jones et al; wavelet analysis of turbulance - the mixed energy cascade, C. Meneneau; hierarchical models of turbulance, P. Frick and V. Zimin; characterization of ATM traffic in the frequency domain, M. Luoni; the self-similarity of d-dimensional potential turbulance, S.N. Gurbatov and A.I. Saichev; solution of Burgers' equation by Fourier transform methods, J. Caldwell; spiral structures in turbulant flow, H.K. Moffatt; fractals in turbulance, J.C. Vassilicos; the physical models and mathematical description of 1/f noise, A. Malakhov and A. Yakimov; fractal models of density interfaces, J.M. Redondo; the fractal dimension of oil-water interfaces in channel flows, G. Saether et al; fractal aggregates in the atmosphere, J.M. Redondo et al; morphology of disordered materials studied by multifractal analysis, J. Muller.