This book could have been entitled "Analysis and Geometry." The authors are addressing the following issue: Is it possible to perform some harmonic analysis on a set? Harmonic analysis on groups has a long tradition. Here we are given a metric set X with a (positive) Borel measure ? and we would like to construct some algorithms which in the classical setting rely on the Fourier transformation. Needless to say, the Fourier transformation does not exist on an arbitrary metric set. This endeavor is not a revolution. It is a continuation of a line of research whichwasinitiated, acenturyago, withtwofundamentalpapersthatIwould like to discuss brie?y. The ?rst paper is the doctoral dissertation of Alfred Haar, which was submitted at to University of Gottingen ] in July 1907. At that time it was known that the Fourier series expansion of a continuous function may diverge at a given point. Haar wanted to know if this phenomenon happens for every 2 orthonormal basis of L 0,1]. He answered this question by constructing an orthonormal basis (today known as the Haar basis) with the property that the expansion (in this basis) of any continuous function uniformly converges to that function."

Five leading specialists reflect on different and complementary approaches to fundamental questions in the study of the Fluid Mechanics and Gas Dynamics equations. Constantin presents the Euler equations of ideal incompressible fluids and discusses the blow-up problem for the Navier-Stokes equations of viscous fluids, describing some of the major mathematical questions of turbulence theory. These questions are connected to the Caffarelli-Kohn-Nirenberg theory of singularities for the incompressible Navier-Stokes equations that is explained in Gallavotti's lectures. Kazhikhov introduces the theory of strong approximation of weak limits via the method of averaging, applied to Navier-Stokes equations. Y. Meyer focuses on several nonlinear evolution equations - in particular Navier-Stokes - and some related unexpected cancellation properties, either imposed on the initial condition, or satisfied by the solution itself, whenever it is localized in space or in time variable. Ukai presents the asymptotic analysis theory of fluid equations. He discusses the Cauchy-Kovalevskaya technique for the Boltzmann-Grad limit of the Newtonian equation, the multi-scale analysis, giving the compressible and incompressible limits of the Boltzmann equation, and the analysis of their initial layers.

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 Poincare 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.

Now in paperback, this remains one of the classic expositions of the theory of wavelets from two of the subject's leading experts. In this volume the theory of paradifferential operators and the Cauchy kernel on Lipschitz curves are discussed with the emphasis firmly on their connection with wavelet bases. Sparse matrix representations of these operators can be given in terms of wavelet bases which have important applications in image processing and numerical analysis. This method is now widely studied and can be used to tackle a wide variety of problems arising in science and engineering. Put simply, this is an essential purchase for anyone researching the theory of wavelets.

Over the last two years, wavelet methods have shown themselves to be of considerable use to harmonic analysts and, in particular, advances have been made concerning their applications. The strength of wavelet methods lies in their ability to describe local phenomena more accurately than a traditional expansion in sines and cosines can. Thus, wavelets are ideal in many fields where an approach to transient behaviour is needed, for example, in considering acoustic or seismic signals, or in image processing. Yves Meyer stands the theory of wavelets firmly upon solid ground by basing his book on the fundamental work of Calderon, Zygmund and their collaborators. For anyone who would like an introduction to wavelets, this book will prove to be a necessary purchase.

This long-awaited update of Meyer's Wavelets: Algorithms & Applications includes completely new chapters on four topics: wavelets and the study of turbulence, wavelets and fractals (which includes an analysis of Riemann's nondifferentiable function), data compression, and wavelets in astronomy. The chapter on data compression was the original motivation for this revised edition, and it contains up-to-date information on the interplay between wavelets and nonlinear approximation. The other chapters have been rewritten with comments, references, historical notes, and new material. Four appendices have been added: a primer on filters, key results (with proofs) about the wavelet transform, a complete discussion of a counterexample to the Marr-Mallat conjecture on zero-crossings, and a brief introduction to Holder and Besov spaces. In addition, all of the figures have been redrawn, and the references have been expanded to a comprehensive list of over 260 entries. The book includes several new results that have not appeared elsewhere. Wavelet analysis-an exciting theory at the intersection of the frontiers of mathematics, science, and technology-is a unifying concept that interprets a large body of scientific research. In addition to its intrinsic mathematical interest, its applications have serious economic implications in the areas of signal and image compression. For these expanding fields, this book provides a clear set of concepts, methods, and algorithms adapted to a variety of applications ranging from the transmission of images on the Internet to theoretical studies in physics. The use of wavelet-based algorithms adopted by the FBI for fingerprint compression and by the Joint Photographic Experts Group for the new JPEG-2000 compression standard confirms the success of this theory. The authors present with equal skill and clarity the mathematical background and major wavelet applications, including the study of turbulence, fractal objects, and the structure of the universe. Never before have the historic origins, the algorithms, and the applications of wavelets been discussed in such scope, providing a unifying presentation accessible to scientists and engineers across all disciplines and levels of training. Written specifically for scientists and engineers with diverse backgrounds, the material is presented in a manner that will appeal to both experts and nonexperts alike. This book is a valuable tool for anyone (from graduate student to expert) faced with signal or image processing problems. It also answers the question, "What are wavelets?" The first seven chapters trace the historical origins of wavelet theory and describe the different time-scale and time-frequency algorithms used today under the term "wavelets." Specific examples include the application of wavelet techniques to FBI fingerprint compression problems and the use of wavelets in the new JPEG standard for still image compression. Applying wavelet analysis methods to signal and image processing, fractals, turbulence, and astronomy is covered in the balance.

Now in paperback, this remains one of the classic expositions of the theory of wavelets from two of the subject's leading experts. This volume discusses the theory of paradifferential operators and the Cauchy kernel on Lipschitz curves with the emphasis firmly on their connection with wavelet bases. Sparse matrix representations of these operators can be given in terms of wavelet bases that have important applications in image processing and numerical analysis. This method is now widely studied and can be used to tackle a wide variety of problems arising in science and engineering. Put simply, this is an essential purchase for anyone researching the theory of wavelets.