Algebraic Numbers and Harmonic Analysis

Science & Math

Yves Meyer

Mathematics

North-Holland Pub. Co.

Physicists and mathematicians are intensely studying fractal sets of fractal curves. Mandelbrot advocated modeling of real-life signals by fractal or multifractal functions. One example is fractional Brownian motion, where large-scale behavior is related to a corresponding infrared divergence. Self-similarities and scaling laws play a key role in this new area. There is a widely accepted belief that wavelet analysis should provide the best available tool to unveil such scaling laws. And orthonormal wavelet bases are the only existing bases which are structurally invariant through dyadic dilations. This book discusses the relevance of wavelet analysis to problems in which self-similarities are important. Among the conclusions drawn are the following: 1) A weak form of self-similarity can be given a simple characterization through size estimates on wavelet coefficients, and 2) Wavelet bases can be tuned in order to provide a sharper characterization of this self-similarity. A pioneer of the wavelet ``saga'', Meyer gives new and as yet unpublished results throughout the book. It is recommended to scientists wishing to apply wavelet analysis to multifractal signal processing.

Wavelets, Vibrations and Scalings (Crm Monograph Series, 9)

American Mathematical Society, Centre de Recherches Mathematiques

<p>Image compression, the Navier-Stokes equations, and detection of gravitational waves are three seemingly unrelated scientific problems that, remarkably, can be studied from one perspective. The notion that unifies the three problems is that of "oscillating patterns", which are present in many natural images, help to explain nonlinear equations, and are pivotal in studying chirps and frequency-modulated signals. The first chapter of this book considers image processing, more precisely algorithms of image compression and denoising. This research is motivated in particular by the new standard for compression of still images known as JPEG-2000. The second chapter has new results on the Navier-Stokes and other nonlinear evolution equations. Frequency-modulated signals and their use in the detection of gravitational waves are covered in the final chapter. In the book, the author describes both what the oscillating patterns are and the mathematics necessary for their analysis. It turns out that this mathematics involves new properties of various Besov-type function spaces and leads to many deep results, including new generalizations of famous Gagliardo-Nirenberg and Poincaré inequalities. This book is based on the "Dean Jacqueline B. Lewis Memorial Lectures" given by the author at Rutgers University. It can be used either as a textbook in studying applications of wavelets to image processing or as a supplementary resource for studying nonlinear evolution equations or frequency-modulated signals. Most of the material in the book did not appear previously in monograph literature.</p>

Oscillating Patterns in Image Processing and Nonlinear Evolution

Evolution

American Mathematical Society

This book is a companion to the first one published by Springer IPTI, on "Wavelets, time frequency methods and phase space". Both books give careful descriptions of the applications of wavelets to relevant problems in scienceand technology. The special flavour of the book is this interaction between conceptual advances ("the wavelet saga" and practial applications (e.g. "monitoring of nuclear plants"). All the speakers who were invited to participate to the conference were truly experts in their respective fields.

Wavelets and Applications: Proceedings of the International Conference Marseille, France, May 1989 (Recherches en Mathematiques Appliquees Research Notes in Applied Mathematics)

Springer

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Books by Yves Meyer