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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.
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