1.1. From the Publisher.

Diophantine approximations to real numbers. Some classical results in Diophantine approximations. Measure-theoretical methods in Diophantine approximations. Diophantine approximations and additive problems in locally compact abelian groups. Uniqueness of representation by trigonometric series. Problems on a-periodic trigonometric sums. Special trigonometric series (complex methods). Special trigonometric series (group-theoretic methods). Pisot numbers and spectral synthesis. Ultra-thin symmetric sets.

1.2. From the Preface.

This book is dedicated to the memory of Raphaël Salem: it contains most of his beautiful discoveries an the proof of his conjecture about the role played by Pisot numbers in the problem of spectral analysis.

1.3. Contents.

Chapter 1: Diophantine approximations to real numbers.

Chapter 2: Diophantine approximations and additive problems in locally compact abelian groups.

Chapter 3: Uniqueness of representation by trigonometric series.

Chapter 4: Problems on a-periodic trigonometric sums.

Chapter 5: Special trigonometric series (complex methods).

Chapter 6: Special trigonometric series (group-theoretic methods).

Chapter 7: Pisot numbers and spectral synthesis.

Chapter 8: Ultra-thin symmetric sets.

1.4. Review by: Henri Joris.
Mathematical Reviews MR0485769 (58 #5579).

A Pisot number is a real algebraic integer $\alpha > 1$, such that $|\alpha′| < 1$ for all its conjugates $\alpha′$. Work done on them since 1938 by Pisot, Salem, Zygmund and others, including the author, has revealed surprising properties of Pisot numbers with respect to Diophantine approximation and to harmonic analysis. They are the examples around which a large part of the theory in the book under review is built, although not appearing in all of the eight chapters. The book is not an introduction to harmonic analysis, but a treatise on selected topics, and includes much of the author's own work. Reading it necessitates some working knowledge of abstract harmonic analysis, functional analysis, distributions and complex functions.

2.1. From the Preface.

The book includes the basic notions that we have tried to present with sufficient motivation and applications to justify the theory: the abstract notion of groups, rings and fields accompany the known concrete examples; the study of polynomials continues with that of divisibility and division with remainder; rational fractions and their decomposition into simple elements are followed by the search for primitives. Finally, we have reserved until the end two chapters on principal ideal domains, morphisms and quotient structures.

2.2. Review by Paolo Maroscia.
Mathematical Reviews MR0434681 (55 #7646).

This is a booklet containing an elementary introduction to general algebra, intended, according to the authors, for future students of mathematics as well as those who will take courses with a predominantly physical or chemical focus. ... There are many examples and exercises with indications of results and solutions.

3.1. From the Preface.

Around the "indefinitely differentiable" theory of pseudo-differential operators (o.p.d.), a "non-orthodox" theory developed, under the leadership of A Calderón and A Zygmund; we are looking for the weakest possible conditions of regularity allowing us to obtain the results that we have, without difficulty, in the usual C∞ cases.

The purpose of these notes is to describe, with precision, this "neighbourhood of classical theory".

We then propose to reduce the hypotheses as much as possible while retaining the desired conclusion. We will most often obtain necessary and sufficient conditions; immediately we leave classical theory (which only describes sufficient conditions).

Naturally this work will force us to introduce demonstrations different from those that can be found in the literature (the latter have often already been pushed to their limits).

Finally, the minimum assumptions to be made will depend on the problem studied.

3.2. Review by: Simon G Gindikin.
Mathematical Reviews MR0518170 (81b:47061).

This book is devoted to results relating to the theory of pseudo-differential operators (p.d.o.): classical p.d.o.'s, Hörmander p.d.o.'s, Calderón-Zygmund operators, etc. There are a number of new results and new proofs of known results. The work consists of six chapters.

4.1. From the Preface.

For a long time the basic functions of analysis were the cosine, the sine and the imaginary exponential. Their study has been and remains an inexhaustible source of problems and discoveries in mathematical analysis. These problems come from the absence of a good dictionary translating the properties of a function into those of its Fourier coefficients. Here is an example of this difficulty. J P Kahane, Y Katznelson and K de Leeuw demonstrated that starting from an arbitrary summable square function $f(x)$, it suffices, to obtain a continuous function $g(x)$, to possibly increase the modulus of the coefficients of Fourier of $f(x)$ and judiciously change their phases. It is therefore impossible to predict the properties (size, regularity) of a function by only knowing the order of magnitude of its Fourier coefficients, it remains difficult to know them explicitly and many problems are still open.

In the early 1980s, many scientists were already using "wavelets" as an alternative to traditional Fourier analysis. This alternative gave hope for a simpler numerical analysis and a more robust synthesis of certain transient phenomena. The "wavelets" of J S Liénard or X Rodet concerned the digital processing of acoustic signals (speech or music) and those of J Morlet were used to store and interpret seismic signals collected during oil prospecting campaigns. On the side of mathematicians, research was also active and, to cite only the main ones, R Coifman and G Weiss created the "atoms" and the "molecules" which were to constitute the basic components of the various functional spaces, the rules of assembly being clearly defined and simple to use. Some of these atomic decompositions could also be obtained by discretising a famous identity, due to A Calderon, and in which "wavelets" figured implicitly. This identity was then found by Morlet and his collaborators ... Finally L Carleson used functions very similar to "wavelets" in order to construct an unconditional basis of the $H^{1}$ space of Stein and Weiss.

These different works had such a "family resemblance" that it was important to group them into a coherent theory, mathematically founded and, at the same time, of universal use. The orthonormal wavelet bases that we learn how to construct in this work replace the empirical "wavelets" of Liénard, Morlet and Roder.

These same orthonormal wavelet bases provide direct access to the "atomic decompositions" of Coifman and Weiss which are thus, and for the first time, linked to the constructions of unconditional bases of usual functional spaces. Wavelet bases are universally used; "whatever comes to hand", whether functions or distributions are the sum of a series of wavelets and, unlike what happens for Fourier series, the coefficients of that series translate in a simple, precise and faithful way the properties of these functions or distributions.

We then have a new instrument allowing us to make, without thinking about it, the delicate constructions that we could previously only carry out using lacunar or random Fourier series; the exceptional properties of the sums of these special series become the banal properties of the generic sums of the wavelet series.

Algorithms for analysing and synthesising orthogonal wavelet series will likely play an important role in various sectors of science and technology and, for all that is known about wavelets, mathematicians, physicists and engineers will find in the first volume of this work.

4.2. Review by: Karlheinz Gröchenig.
Mathematical Reviews MR1085487 (93i:42002).

The official birth of wavelet theory occurred in 1986 with the discovery of a smooth function ψ, now called a "wavelet''... Retrospectively, wavelet theory emerged simultaneously and independently in several fields: under the name of subband coding in image processing, in quantum field theory (G Battle, P Federbush), in approximation theory where J O Strömberg actually had constructed the first wavelet basis in 1982, and as part of Calderòn's program pursued by Meyer and his school. Since then wavelet theory has attracted much attention and has become a very active field of research.

The author's monograph in three volumes is the first comprehensive account of wavelet theory. Wavelets are not pursued as an end in themselves, but as a new and extremely useful tool for Fourier analysis. Thus the three volumes contain new and elegant approaches to topics in approximation theory, the theory of Hardy spaces or the theory of singular integral operators. The development of the tool takes up only half of Volume I, whereas the rest is concerned with the mathematical applications of wavelets. The first volume is devoted to multiresolution analysis, wavelet bases and the theory of classical function spaces, and addresses a wider audience of mathematicians, physicists, and engineers.

5.1. From the Publisher.

Both in pure and applied mathematics and in signal and image processing, a new mathematical theory tends to replace, in certain cases, traditional Fourier analysis. This concerns certain sectors of science and technology such as the study of turbulence, musical acoustics, etc.

The recent mathematical theory of wavelets, coherent and fruitful, opens a new field for analysis.

This second volume is the natural continuation of the first. But it can also be approached directly and the reader will find there a clear and complete presentation of the work of Calderon and Zygmund in operator theory, work which was the source of very great discoveries in complex analysis, in linear partial differential equations and non-linear partial differential equations, etc. The "wavelet enthusiast" will have the pleasure of finding what is required when it comes to obtaining efficient matrix representations of large classes of Calderon-Zygmund operators.

The author is a professor at the Centre de Recherche en Mathématiques de la Décision at Paris-Dauphine University and a corresponding member of the Academy of Sciences; he was previously responsible for teaching at the mathematics centre of the École Polytechnique.

5.2. Review by: Karlheinz Gröchenig.
Mathematical Reviews MR1085488 (93i:42003).

The second volume is a beautiful introduction to the theory of Calderón-Zygmund operators and is written mostly for the mathematical analyst. Volume II is essentially self-contained and can be studied independently from Volume I. Only the existence of smooth orthonormal wavelet bases and certain characterisations of function spaces are required from Volume I.
...
The author's books offer the most profound and comprehensive treatment of wavelets. Wavelet theory is seen as an important development in harmonic analysis, which deserves a prominent place in a long and rich tradition. It is to be hoped that this outstanding and inspiring book will appear soon in an English translation so that it will also be accessible to an even larger audience.

6.1. From the Publisher.

Without a doubt, non-linear analysis will be at the heart of science in the coming century. One of the paths of access, proposed by A Calderon, is the systematic study of multilinear operators. The examples presented in volume III range from complex analysis to non-linear partial differential equations and, in each of the case, the corresponding multilinear operators can be studied effortlessly thanks to wavelets and the results obtained in volumes I and II of this work.

6.2. Review by: Gerald B Folland.
Mathematical Reviews MR1160989 (93i:42004).

This volume is the third and final part of a treatise on a circle of recent developments in analysis that includes the theory of wavelets and the "Calderón program" of analysis of singular integrals with minimal smoothness hypotheses. The first two volumes were devoted respectively to wavelets and to the generalised singular integrals that have come to be known as Calderón-Zygmund operators. In the present volume, the focus is on applications of the Calderón program to complex analysis and nonlinear problems in differential equations; wavelets have receded into the background and make only a brief appearance in the final section.
...
This volume, as well as its predecessors, contains a wealth of recently developed material for which a unified exposition has not been previously available, and it is written in a discursive and conversational style that generally makes for easy reading. For these reasons it is a most valuable addition to the literature.

7.1. From the Publisher.

Wavelet analysis has kept its promises. This discipline has truly exploded, thanks to very active collaboration between physicists experienced in the methods of aquatic mechanics and experts in signal and image processing. The introduction of oblique wavelets complementing the orthonormal wavelet bases allowed the diversification of analysis techniques. Alongside these tools, which constitute time-scale methods, the book also presents time-frequency methods; these are used in situations where wavelet-based algorithms are not suitable. The expected applications of these new algorithms concern speech signals. Thus, image and speech are analysed using new methods that do not appear in other treatises.

7.2. Review by: Ewald Quak.
Mathematical Reviews MR1204654 (94g:42059).

These lecture notes are based on graduate-level courses taught in France and the United States. According to the introduction, the author wants to address some topics not covered in a previous monograph of his [Ondelettes et operateurs. I, II, III]. He starts out in Chapter 1 by briefly covering Fourier transforms for functions in $L^{1}(\mathbb{R})$ and $L^{2}(\mathbb{R})$ and for tempered distributions as well as Fourier series and sampling theorems for functions of compact support and band-limited functions. Discrete transforms, i.e. the discrete Fourier transform, the fast Fourier transform and the discrete cosine transform, are introduced in Chapter 2.

The Malvar wavelets are discussed in Chapter 3. It is shown that their discrete versions form an orthonormal basis of $<curlyl>^{2}(\mathbb{Z})$ and that the continuous versions yield an orthonormal basis of $L^{2}(\mathbb{R})$. This is followed by a discussion of the orthogonal projections associated with the Malvar wavelets.
...
Chapter 5 starts with the definition of a multiresolution analysis (MRA) and some examples. It is then investigated how nonorthogonal projections and their adjoints can be used to generate dual Riesz bases and, consequently, two MRAs that are dual to each other. Chapter 6 covers the pyramid schemes of Burt and Adelson and discusses how they fit into the framework of dual MRAs, focussing especially on one example due to Burt and Adelson. In Chapter 7, as a specific example, a pair of dual Riesz bases is constructed in detail, starting with one compactly supported piecewise linear function. Chapter 8 reviews the constructions of orthonormal wavelets by the author, Stromberg, Battle and Lemarie, and Daubechies. Some final remarks conclude these notes.

8.1. Note.

This is the English translation of 4. above. In the preface to this edition, the author praises the quality of the translation.

8.2. Review by: Charles K Chui.
Bulletin of the American Mathematical Society 33 (1) (1996), 131-134.

Although wavelet analysis is a relatively young mathematical subject, it has already drawn a great deal of attention, not only among mathematicians themselves, but from various other disciplines as well. In fact, it is fair to attribute the main driving force of the rapid development of this field to the "users" rather than to the "inventors" of mathematics. To the mathematicians, Fourier analysis has been and still is a very important research area. Its theory is beautiful, its techniques powerful, and its impact on science and technology most profound. However, even as early as the decade of the 1940s, those who used the Fourier approach to analyse natural behaviours were already frustrated with the limitation of the Fourier transform and Fourier series in the investigation of physical phenomena with nonperiodic behaviour and local variations. The need for simultaneous time-frequency analysis led to the introduction of Gabor's short-time Fourier transform in 1946 and the so-called Wigner-Ville transform in 1947. But the common ingredient of these two transforms is the sinusoidal kernel in the core of their definitions, so that both high- and low-frequency behaviours are investigated in the same manner and any signal under investigation is matched by the same rigid sinusoidal waveform. In place of the sinusoidal kernel as modulation (for phase shift), a French geophysicist, J Morlet, introduced in 1982 the operation of dilation, while keeping the translation operation, and developed an algorithm for the recovery of the signals under investigation from this "wavelet transform". It was the mathematical physics group in Marseille, led by A Grossmann, in cooperation with I Daubechies, T Paul, etc., that extended Morlet's discrete version of wavelet transform to the continuous version, by relating it to the theory of coherent states in quantum physics. This was how the notion of the integral (or continuous) wavelet transform was introduced.

The development of the mathematical analysis of the wavelet transform had really not begun, until a year later, in 1985, when Yves Meyer learnt about the work of Morlet and the Marseille group and immediately recognised the connection of Morlet's algorithm to the notion of resolution of identity in harmonic analysis due to A Calderón in 1964. He then applied the Littlewood-Paley theory to the study of "wavelet decomposition". In this regard, Yves Meyer may be considered as the founder of this mathematical subject, which we call wavelet analysis. Of course, Meyer's profound contribution to wavelet analysis is much more than being a pioneer of this new mathematical field. For the past ten years, he has been totally committed to its development, not only by building the mathematical foundation, but also by actively promoting the field as an interdisciplinary area of research. The book Wavelets and operators is the English translation of his first monograph on this subject. In addition to the two subsequent volumes in this three-volume series (the last jointly with R Coifman), he wrote at least two other shorter monographs on the theory, algorithms, and applications of this subject. D H Salinger should be congratulated for an excellent job in translating this classic volume from French to English.
...
In summary, Meyer's book is the English translation of the first book on an introduction to the mathematical analysis of wavelets. Based on classical Fourier analysis, the author gives a detailed treatment of the construction of wavelets and the application of wavelet series representations to the analysis of the most important function spaces. It is a summary of the development of wavelet analysis in the 1980s and is a classic of this fast-developing field of mathematical analysis.

8.3. Review by: Ingrid Daubechies.
Science, New Series 262 (5139) (1993), 1589-1591.

The book under review was an instant classic when it came out in French. I know at least one mathematician who actually studied French so as to be able to read Ondelettes et Operateurs; many more have been eagerly awaiting the publication of Wavelets and Operators. The translation covers the first volume of the three-volume French work (the other two volumes are the "et operateurs" part, dealing with Calderon-Zygmund operators in their most recent and broadest incarnation and related developments).

The book starts with an introductory chapter about Fourier series and integrals, pointing to wavelet precursors in the mathematical literature, and then proceeds to a discussion of multiresolution analysis (chapter 2) and the corresponding orthonormal wavelet bases in one and more dimensions (chapter 3). Chapter 4 contains a short discussion of nonorthogonal and redundant wavelet families in L2(R) (frames). The last two chapters contain a (sometimes terse) discussion of the many functional spaces where wavelets provide an unconditional basis. I recommend this book to every mathematically minded reader interested in wavelets; it is beautifully written, and the English translation is excellent, staying close to the original without feeling awkward. I found very few typos, and I like the typography better than that of the original-theorems are indicated in boldface now and so are easier to spot when leafing through the book. The reference list has been extended slightly and some references to works that had not yet been published when the French edition came out have been updated. The addition of an index is very useful.

8.4. Review by: D H Griffel.
The Mathematical Gazette 79 (484) (1995), 227-228.

In some ways this book is similar to Ingrid Daubechies' Ten lectures on wavelets ... . They are both good introductions to wavelet theory by distinguished workers in the field. Daubechies is a longer and richer book, with many examples and a good deal of explanation in intuitive terms. Meyer focuses much more on the general theory, and is writing for mathematicians with a good knowledge of analysis; on page 2 the reader is assumed to be familiar with distributions as members of the dual of the space of smooth functions of compact support. On page 14 we read "if the well-known characterisation of Hardy's $H^{p}$ spaces in terms of Lusin's area function is written in the language of atomic decompositions due to Coifman and Weiss, then ...". Meyer does then tell you what Hardy spaces are, and describes Lusin's work; but everything is brief, elegant, and sophisticated.

Wavelets and operators is designed for readers who are thoroughly at home with modem mathematical analysis, function spaces and all that, and who want to focus on the theory rather than examples and applications. It is very well written, and says just enough to make the meaning clear, but no more. Mathematicians interested in wavelets are fortunate in having two such excellent and complementary texts as Meyer and Daubechies available.

9.1. From the Preface.

The "theory of wavelets" stands at the intersection of the frontiers of mathematics, scientific computing, and signal processing. Its goal is to provide a coherent set of concepts, methods, and algorithms that are adapted to a variety of nonstationary signals and that are also suitable for numerical signal processing.

This book results from a series of lectures that Mr Miguel Artola Gallego, Director of the Spanish Institute, invited me to give on wavelets and their applications. I have tried to fulfil, in the following pages, the objective the Spanish Institute set for me: to present to a scientific audience coming from different disciplines, the prospects that wavelets offer for signal and image processing.

A description of the different algorithms used today under the name "wavelets" (Chapters 2-7) will be followed by an analysis of several applications of these methods: to numerical image processing (Chapter 8), to fractals (Chapter 9), to turbulence (Chapter 10), and to astronomy (Chapter 11). This will take me out of my domain; as a result, the last two chapters are merely resumes of the original articles on which they are based.

I wish to thank the Spanish Institute for its generous hospitality as well as its Director for his warm welcome. Additionally, I note the excellent organisation by Mr Perdo Corpas.

My thanks go also to my Spanish friends and colleagues who took the time to attend these lectures.

9.2. Review by: M Victor Wickerhauser.
SIAM Review 36 (3) (1994), 526-528.

The book travels an enormous distance in a slim 133 pages. It describes the construction and the properties of various kinds of wavelets, including Grossmann-Morlet "continuous" wavelets, Daubechies "discrete" orthogonal wavelets, Gabor-Malvar wavelets, and libraries of wavelet packets and other time-frequency atoms. It places these objects in the context of many traditional and new algorithms: compression, analysis of signals in time and frequency, and feature detection through decomposition into basic units. At the end of the book there are four short chapters on vision, fractals, turbulence, and cosmology, where some wavelet-like decompositions have recently yielded new results.

In effect Meyer develops the notion of a meta-algorithm based on choosing the appropriate decomposition units for a particular problem. His book is a brief encyclopaedia of the functions that have recently become available, and gives some of the formulas essential to using them in particular problems. It is not a programmer's guide, but is rather an algorithm developer's guide, useful to the supervisor who must decide how to decompose in order to best reveal the features of interest or calculate the needed quantity.

Meyer writes in a fine colloquial style that is preserved in Ryan's translation. There are references to Montaigne, Roland Barthes, and Mandelbrot's interview in France-Culture. In one chapter, Meyer writes how "with keen pleasure" he reread Galand's thesis introducing quadrature mirror filters in 1983. In another, he hopes that "readers who are mathematicians will enjoy reading this chapter as much as we have enjoyed writing it." The enjoyment that Meyer felt makes the text much more dynamic than, for example, a typical recent journal article on wavelets and
applications.

The main charm of the text is that it uses mathematical rigour both to delineate where the new algorithms are valid, and also to indicate what we may expect if we go out of bounds. Thus, Meyer gives a counterexample in Chapter 8 showing that Mallat's reconstruction from edges does not always recover the exact image, but then explains "why Mallat's algorithm works in practice with such excellent precision": step functions are correctly recovered, and "the signals in question have more in common with step functions than with the subtle functions described in the counterexamples." In Chapter 7 he describes a "catastrophic phenomenon": Seré's result that Gabor wavelets can be arbitrarily separated in the idealised time-frequency plane yet still have enough overlap so that their superposition has infinite energy. Then he points out the root cause (arbitrary eccentricity of the time-frequency support rectangle) and describes various ways around the problem (Malvar wavelets and wavelet packets). He does not neglect to mention the advantages and disadvantages of either alternative, and includes a consideration of the computational costs involved.

9.3. Review by: Ingrid Daubechies.
Science, New Series 262 (5139) (1993), 1589-1591.

Wavelets: Algorithms and Applications is meant for a much larger audience [than the previous work]. The book grew out of a series of lectures presented to a varied group of scientists at the Spanish Institute in Madrid in 1991. Chapters 1 through 7 cover signal analysis, mathematics and vision, the use of subband filtering schemes to build wavelets, and different types of time-frequency analysis, with relevant historical background material. (One chapter discusses "Malvar's wavelets," which are not really wavelets at all but rather a new and much better version of the windowed Fourier transform that is due to Ronald Coifman and Meyer. Because a construction having some of its properties - though not the most interesting, the split-and-merge algorithm - had already been proposed by Henrique Malvar, Meyer, with typical modesty, named these functions Malvar's wavelets.) The last four chapters discuss some applications of wavelets to image analysis, fractals, turbulence, and astronomy, mentioning work in progress and referring the reader to the literature.

In many respects the book is a personal view of the field, and others would no doubt have tackled the undertaking quite differently; this actually adds to the book's interest. There are still mathematical formulas on almost every page, so that I would not recommend the book to mathophobes, but I believe it is accessible to any scientifically minded reader with rudimentary knowledge of Fourier analysis; furthermore, the seasoned mathematician will find discussions of many interesting non-mathematical topics, so that the book is by no means superfluous for readers of the other book under review here. The translation is slightly less close to the original lecture notes, but that can be considered a plus, since a very close translation would not have captured the casual style of the notes. Some of the material has been revised and updated by the translator, Robert D Ryan, in collaboration with Meyer. I noticed a few typos, and I guess Ryan is not a musician, since on page 6 of the English translation the reader is still told that "ré mineur" is a musical note. But these are minor blemishes, and I recommend the book as a delightful introduction to wavelets.

9.4. Review by: Junjiang Lei.
Mathematics of Computation 63 (208) (1994), 822-823.

This is an important book on wavelet analysis and its applications, written by one of the pioneers in the field. It is based on a series of lectures given in 1991 at the Spanish Institute in Madrid. The text was revised and translated in admirable fashion by Robert D Ryan. The book presents recent research on wavelets as well as extensive historical commentary. The mathematical foundations of wavelet theory are dealt with at length, but not to the exclusion of relevant applications. Signal processing is especially emphasised, it being viewed here as the source from which wavelet theory arises. The text is well written in a clear, vivid style that will be appealing to mathematicians and engineers.

The first chapter gives an outline of wavelet analysis, a review of signal processing, and a good glimpse of the contents of subsequent chapters. Chapter 2 sketches the development of wavelet analysis (which can be traced back to Haar and even to Fourier). Here we find explanations of time-scale algorithms and time-frequency algorithms, and their interconnections. Chapter 3 begins with remarks about Galand's work on quadrature mirror filters, which was motivated by the possibility of improving techniques for coding sampled speech. The author then leads us to the point where wavelet analysis naturally enters, and continues to an important result on convergence of wavelets and an outline of the construction of a new "special function" - the Daubechies wavelet. Further discussion of time-scale analysis occupies Chapter 4. In this chapter the author uses the pyramid algorithms of Burt and Adelson in image processing to introduce the fundamental idea of representing an image by a graph-theoretic tree. This provides a background for some of the main issues of wavelet analysis, such as multiresolution analysis and the orthogonal and bi-orthogonal wavelets, that are the main topics of this chapter. From Chapters 3 and 4 readers can see how quadrature mirror filters, pyramid algorithms, and orthonormal bases are all miraculously interconnected by Mallat's multiresolution analysis. Chapters 5 through 7 are devoted mainly to time-frequency analysis. In Chapter 5, Gabor time-frequency atoms and Wigner-Ville transforms are viewed from the perspective of wavelet analysis. Not only is this of independent interest, but it also motivates the next two chapters as well. Chapter 6 discusses Malvar wavelets, especially the modification due to Coifman and Meyer that allows the wavelets to have windows of variable lengths. An adaptive algorithm for finding the optimal Malvar basis is then described. Chapter 7 concentrates on wavelet packets and splitting algorithms. These algorithms are useful in choosing an optimal basis formed by wavelet packets. Borrowing the words of Ville (1947), the author emphasises the following points in Chapters 6 and 7: In the approach of Malvar's wavelets, we "cut the signal into slices (in time) with a switch; then pass these different slices through a system of filters to analyse them." In the approach of wavelet packets, we "first filter different frequency bands; then cut these bands into slices (in time) to study their energy variations."

The last four chapters introduce some fascinating and promising applications of wavelets. The first of these is Marr's analysis of the processing of luminous information by retinal cells. In particular, Marr's conjecture and a more precise version of it due to Mallat are discussed. Marr's conjecture concerns the reconstruction of a two-dimensional image from zero-crossings of a function obtained by properly filtering the image. In Chapter 9 it is shown that, for some signals, wavelet analysis can reveal a multifractal structure that is not disclosed by Fourier analysis. To this end, two famous examples, the Weierstrass and Riemann functions (which show that a continuous function need not have a derivative anywhere), are examined from the viewpoint of wavelet theory. Chapters 10 and 11 describe how wavelets can shed new light on the multifractal structure of turbulence and on the hierarchical organisation of distant galaxies

9.5. Review by: Peter A McCoy.
Mathematical Reviews MR1219953 (95f:94005).

The study of wavelets and their applications has enjoyed considerable attention over the past decade because of their utility in addressing problems in a wide range of disciplines. Among these areas one finds signal analysis, image processing, scientific computing, turbulence and astronomy. Yves Meyer, who is one of the principals in the development of this discipline, has written a particularly fine monograph, in conjunction with Ryan as translator, which serves as a highly "readable" and worthwhile introduction to the subject. The purpose of the book is to describe certain coding algorithms that have recently been used to analyse signals with a fractal structure or for the compression and storage of information. Having said this, I will summarise the contents and features of the book.

The first chapter gives the reader an overview of the scientific content of the book, acquaints the reader with the notions of a signal and signal processing, and introduces the main objectives and issues of the study. Although the author notes many areas of application, ranging from telecommunications and image processing to neurophysiology, cryptography and archived data found in the FBI's fingerprint records, I would view the theme in the context of signal and image processing, where the relevant issues are analysis and diagnostics, coding, quantisation and compression, transmission, storage and synthesis and reconstruction. Some basic terminology and concepts for "time-frequency" (Grossmann-Morlet) and "time-scale" (Gabor and Malvar) type wavelets are set up along with the framework for the development of algorithms.

The second chapter has a special status. It develops a historical perspective of wavelets and frames some of the major topics for discussion in the following chapters. It seems that wavelets are synthesised from roughly seven related concepts. The idea begins with Fourier (1807), who represented a function in terms of its constituent frequencies. Fourier analysis is inadequate in describing many phenomena that have a random, "rough" or fractal structure. The first steps toward an attempt to address functions with these characteristics were taken by Paul Du Bois-Reymond (1873) and Haar (1909), in whose studies Fourier's frequency analysis became an analysis of scale.
...
The reader will find a nice balance between the detail of "hard" analysis and the exposition supporting the development of the concepts and applications. The detail is at a level appropriate to support the analysis, yet is not too detailed to burden the "nonexpert" reader who seeks to evaluate the potential for application of wavelets in a particular area of research. There are a number of nice illustrations that explain the algorithms and analysis in "pictorial" form. The monograph is the result of the efforts of an author and a translator with insight and ability.

10.1. From the Publisher.

Currently, new trends in mathematics are emerging from the fruitful interaction between signal processing, image processing, and classical analysis.

One example is given by "wavelets", which incorporate both the know-how of the Calderon-Zygmund school and the efficiency of some fast algorithms developed in signal processing (quadrature mirror filters and pyramidal algorithms.)

A second example is "multi-fractal analysis". The initial motivation was the study of fully developed turbulence and the introduction by Frisch and Parisi of the multi-fractal spectrum. Multi-fractal analysis provides a deeper insight into many classical functions in mathematics.

A third example - "chirps" - is studied in this book. Chirps are used in modern radar or sonar technology. Once given a precise mathematical definition, chirps constitute a powerful tool in classical analysis.

In this book, wavelet analysis is related to the 2-microlocal spaces discovered by J M Bony. The authors then prove that a wavelet based multi-fractal analysis leads to a remarkable improvement of Sobolev embedding theorem. In addition, they show that chirps were hidden in a celebrated Riemann series.

Features:

Provides the reader with some basic training in new lines of research.

Clarifies the relationship between point-wise behaviour and size properties of wavelet coefficients.

Readership

Graduate students and researchers in mathematics, physics, and engineering who are interested in wavelets.

10.2. Contents.

Introduction

I. Modulus of continuity and two-microlocalization.

II. Singularities of functions in Sobolev spaces.

III. Wavelets and lacunary trigonometric series.

IV. Properties of chirp expansions.

V. Trigonometric chirps.

VI. Logarithmic chirps.

VII. The Riemann series.

10.3. Abstract.

We investigate several topics related to the local behaviour of functions: pointwise Hölder regularity, local scaling invariance and very oscillatory "chirp-like" behaviours. Our main tool is to relate these notions to two-microlocal conditions which are defined either on the Littlewood-Paley decomposition or on the wavelet transform. We give characterisations and the main properties of these two-microlocal spaces and we give several applications, such as bounds on the dimension of the set of Hölder singularities of a function, Sobolev regularity of trace functions, and chirp expansions of specific functions.

10.4. Review by: Boris S Rubin.
Mathematical Reviews MR1342019 (97d:42005).

The monograph is devoted to investigation of the local behaviour of functions (pointwise regularity, local sealing invariance, strong oscillation, etc.). As a main tool the authors use the notion of two-microlocalisation, which is based on the Littlewood-Paley decomposition or on the wavelet transform. The characterisations of two-microlocal spaces are given and the main properties of such spaces are described. As an application of these results the authors consider bounds on the dimension of the set of Hölder singularities of a function, Sobolev regularity of trace functions, and chirp expansions.

11.1. From the Publisher.

Physicists and mathematicians are intensely studying fractal sets of fractal curves. Mandelbrot advocated modelling of real-life signals by fractal or multifractal functions. One example is fractional Brownian motion, where large-scale behaviour 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 characterisation through size estimates on wavelet coefficients, and 2) Wavelet bases can be tuned in order to provide a sharper characterisation 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.

Titles in this series are co-published with the Centre de recherches mathématiques.

Readership

Graduate students, research mathematicians, physicists, and other scientists working in wavelet analysis.

11.2. From the Preface.

Multifractal analysis makes it possible to discover hidden laws in the apparent irregularity of certain functions or certain signals. This concerns developed turbulence, many stochastic processes and even the study of the human genome.

Multifractal analysis also reveals the fine structure of some of the irregular functions constructed by Gilles Deslauriers and Serge Dubuc.

Winner of the Aisenstadt chair at the Mathematical Research Center of the University of Montreal, I had a double obligation: to give five lessons on a theme of my choice, then to write a book which completes and extends these presentations.

The theme chosen for these lessons was multifractal analysis, as presented in the beautiful book by Alain Arnéodo and in the remarkable work of Stephane Jaffard.

But I still had to write a book.

Does wavelet analysis make it possible, as Alain Arnéodo states, to "dive into the very heart of multifractal structures"?

This question has always intrigued me and my book is the beginning of an answer.

I study the relevance of spectral methods (wavelets or Littlewood-Paley analysis) in various aspects of multifractal analysis.

Vibrations, wavelets and scaling laws, this is the subject of this book.

The following treatise therefore does not correspond exactly to the five lessons I gave. Nor is it an introduction to multifractal analysis. This text should rather be used as a rigorous and severe clarification, completing the work of Arnéodo and Jaffard.

I would like to particularly thank the Mathematical Research Center of the University of Montreal as well as the organisers of the "wavelet program". The very happy week I spent in Montreal was the starting point for this work.

Isabelle Paisant, who typed all my works on wavelets, once again agreed to help me. Without his collaboration, I would not have succeeded in completing this essay, which has been restarted so many times.

11.3. Review by: James S Walker.
Bulletin of the London Mathematical Society 31 (1999), 740-743.

Meyer discusses several deep results in function theory on Rn, where the roots of wavelet analysis lie. ...The first half of the book describes some of Meyer's new results on wavelet-based characterisations of local Hölder exponents, and other quantities which Meyer calls scaling exponents. ... The book concludes with two chapters that describe a new type of wavelet bases. The elements of these bases have their supports and the supports of their Fourier transforms essentially concentrated in rectangles that partition the time-frequency plane in a manner which provides a much finer division near the zero-frequency axis than is the case for standard wavelet bases. ... In addition to these general theoretical results, Meyer's book is also sprinkled throughout with a fascinating collection of examples and counter-examples. ... Meyer's book is suitable for professional researchers in function theory, or as a text for an advanced graduate seminar. It contains no exercises, but Meyer writes in a compressed yet lucid style which invites the reader's participation.

11.4. Review by: Lars F Villemoes.
Mathematical Reviews MR1483896 (99i:42051).

This monograph grew out of five lectures given by the author at the University of Montreal on the theme of multifractal analysis. It exposes and completes the work of S Jaffard and the author on pointwise regularity and local oscillations of functions, and several mathematical aspects of the recent work of A Arnéodo on multifractals are studied.

12.1. From the Publisher.

This long-awaited update of Meyer's Wavelets: Algorithms and 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 Hölder 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.

12.2. From the Preface.

Wavelet analysis is a branch of applied mathematics that has produced a collection of tools designed to process certain signals and images. This new book is devoted to describing some of those tools, their applications. and their history.

We will trace several of the technical roots of wavelet analysis, going back to the 1930s and before. These are examples of where the mathematical techniques that we now codify as wavelet analysis first appeared. They are for the most part concerned with the internal structure of mathematics itself. We judge that the applied point of view began after World War II and was embedded in a more general philosophical context exemplified by an ambitious program called The Institute for the Unity of Science. This "institute without walls" was a vision, a vision that was shared by such prominent scientists as John von Neumann, Claude Shannon, and Norbert Wiener. It was the time when Claude Shannon discovered the laws that govern the coding and transmission of signals and images. It was the time when Norbert Wiener and John von Neumann unveiled the relationships between mathematical logic, electronics, and neurophysiology. This led to the design of the first computers. It was the time when Dennis Gabor proposed that speech signals should be decomposed into a series of time-frequency atoms he named "logons." It was the time when Eugene P Wigner and Léon Brillouin introduced the time-frequency plane.

These pioneering scientists opened new avenues in science, and one of these avenues is called time-frequency analysis. Time-frequency analysis, which is based on Gabor wavelets, will be one of the main topics of this book. Gabor wavelets were improved by Kenneth Wilson, Henrique Malvar, and finally by Ingrid Daubechies, Stéphane Jaffard, and Jean-Lin Journé.

In contrast with this established line of research, time-scale analysis has had a harder time. Indeed, time-frequency analysis yields the musical score, the notes with their frequencies and durations of the music we hear. Time-scale analysis focuses on the transients, the attack of the trumpet, which lasts a few milliseconds, and similar nonstationary signals. While time-frequency analysis was born in the 1940s, time-scale analysis emerged in the late 1970s in completely distinct areas such as image processing (E H Adelson and P J Burt), neurophysiology (David Marr), quantum field theory (Roland Seneor, Jacques Magnen, Guy Battle, Paul Federbush, James Glimm, and Arthur Jaffe), and in geophysics (Jean Morlet). The outstanding collaboration between Alex Grossmann and Jean Morlet gave birth to a new vision that emerged in the 1980s, and the message was the following: While stationary or quasi-stationary signals are adequately decomposed into a series of time-frequency atoms or Gabor-like wavelets, signals with strong transients are better analysed with the time-scale wavelets developed by Grossmann and Morlet.

A spectacular example where time-frequency analysis and time-scale analysis have been able to compete is the new JPEG-2000 compression standard for still images. This new standard is based all time-scale wavelets. The old JPEG standard was based on an algorithm called the discrete cosine transform, which is a kind of windowed Fourier transform. This algorithm belongs to the time-frequency group. (Here one ought to say "space-frequency," since an image is a two-dimensional signal.) In the case of JPEG-2000, and in similar compression problems, time-scale wavelets have been preferred over time-frequency wavelets. This success story was not available when the original book first appeared.

This new book began as a one-chapter revision of Wavelats: Algorithms & Applications (SIAM, 1993), which is based on lectures Yves Meyer delivered at the Spanish Institute in Madrid in February 1991. While Yves Meyer and Robert Ryan were working on the translation and revision of the new chapter, which ultimately became Chapter 11 of the current book, it became clear, based on the many developments in both the theory and applications since 1993, that an extensive revision of the original book was needed. Since Stephane Jaffard already had suggested a number of changes and additions particularly in the sections involving the analysis of multifractal functions, where he is a recognised expert, he was invited to join the project. The result of our collaboration is an almost completely new book, and thus we have given it a new title. Although we have retained the core of the first four chapters, many parts of these chapters have been rewritten and expanded, particularly Chapters 1 and 2. Appendix A has been added as an introduction to some basic filter concepts and hence as a complement to Chapter 3. Chapter 5 has been completely rewritten: it contains new material on chirps that was not known when the first edition was published. Chapters 6 and 7 have been slightly expanded, but they generally follow the original texts. Rather than expanding Chapter 8, we have added Appendix C, which is devoted to a complete discussion of a counterexample to a conjecture of Stephane Mullat on zero-crossings. This counterexample was outlined in the first edition, but this is the first time the details have been published.

Chapters 9 and 10, although based on the first edition, are considerably expanded and hence essentially new. Chapter 9 (formerly Chapter 10) tells a much more complete and up-to-date story about the use of wavelets for the study of turbulence. Chapter 10 (based on the former Chapter 9) contains a complete analysis of the Weierstrass and Riemann functions, plus a general discussion about the use of wavelets to analyse multifractal functions. Appendix B complements Chapter 10 by providing key results (with proofs) about some wavelet transforms and their inverses. The treatment here is perhaps slightly different from other developments of this now-classical theory.

Chapter 11 is the original motivation for this new book, and we consider it the centrepiece. Here we discuss the intriguing interaction between wavelets and nonlinear analysis and the applications of this line of research to image compression and denoising. Since this chapter involves the concepts that may not be familiar to some readers, we have added Appendix D to introduce Hölder and Besov spaces, plus results on their characterisations in terms of wavelet coefficients.

The original edition contained two pages about the then-emerging use of wavelets in astronomy. It was written at a time when the applications of wavelets to astronomy were received with scepticism. Wavelets are today recognised as an essential tool in astronomy. This story has been expanded in Chapter 12, where we have written a detailed analysis of how wavelets are used in two specific algorithms. We also discuss the use of wavelets to understand the hierarchical structure of the universe and its evolution. This is embedded in a historical context going back to the eighteenth century.

The bibliography has been considerably expanded to include research papers from each of the applications discussed, as well as many books and papers of general or historical interest. We have not listed any of the many websites that exist. Instead, we encourage the reader to visit the "official" wavelet site, www.wavelet.org, which is edited by Wim Sweldens with support from Lucent Technologies. Here one will find lists of regularly updated references, a calendar of events, links to homepages of researchers, and links to sites from which wavelet software can be downloaded.

12.3. Review by: Peter A McCoy.
Mathematical Reviews MR1827998 (2002g:00007).

The authors have written a new, up-to-date, readable book that reflects the many advances in the science of "wavelets" since the publication in 1993 of [Y F Meyer, Wavelets, Translated from the French and with a foreword by Robert D Ryan, 1993]. Although portions of the current book retain some of the central features of the original, the material has been rewritten and enhanced with information that was unknown in 1993. The bibliography has been greatly expanded to include research papers pertinent to each of the applications discussed, as well as books and papers of a general or historical interest. The thrust of the book is on the processing of one- and two-dimensional signals, the latter of which form images. The main tool is the Gabor-wavelet-based time-frequency analysis. The centrepiece of the book is the interaction between wavelets, nonlinear analysis and their applications to image compression and denoising as discussed in Chapter 11.

The first chapter provides a compact overview of the content of the book. The notions of signal and image processing are introduced, including their terminology and goals. The topics presented include stationary and adaptive signals and adaptive coding. Short overviews are given of the Grossman-Morlet time-scale and the Gabor and Malvar and Wilson time-frequency wavelets, as well as optimal algorithms and optimal representation. The authors note that their purpose is to describe a group of coding algorithms that have been shown over the past few years to be especially effective for compression and analysing certain nonstationary signals.
...
The bibliography has been greatly expanded since the first book to include research papers from each of the applications discussed and references of historical interest. The subject and author indices are complete and easy to use. The diagrams and artwork are excellent. The book provides a superb overview of "wavelets".

12.4. Review by: M Victor Wickerhauser.
SIAM Review 44 (2) (2002), 302-305.

The preface of Wavelets: Tools for Science & Technology states that Chapter 11 was the authors' original reason to revise, but it is clear that a decade of research has changed the way the rest of the material has to be presented. There are 264 items in the new book's bibliography, plus an extensive author index, compared with 86 (an average of 8 per chapter, with some repetitions) scattered about the earlier book. The new bibliography has unusual backward pointers, giving the page numbers where citations appear in the text, so that with a little work it is still possible to browse by chapter or subject.

Some chapters are strictly augmented with material that was not known in 1992. For example, orthonormal modulated Malvar wavelets, or "chirplets," appeared in 1996 and are in the new Chapter 6. A very important existence result for Navier-Stokes flows, which was discovered by Meyer's student Cannone in 1995, appears in the new Chapter 10. Wavelet transforms to repair Hubble Space Telescope images, which were superseded by a hardware fix in late 1993, became standard astronomical image enhancement tools by 1997 and are now discussed in a vastly expanded Chapter 12.

The four new appendices cover mathematical technicalities that, while important, would derail the narrative if discussed in the main text. For example, Appendix A on quadrature filter fundamentals contains a complete proof that every continuous filter operator on finite-energy sequences is determined by the Fourier coefficients of some bounded periodic transfer function. None of the difficulties addressed in that proof arise in the finite impulse response (FIR) filter case, which includes all compactly supported wavelets and most of the applications. Likewise, the complete details of a counterexample to Mallat's conjecture about perfect reconstruction from wavelet maxima are placed in Appendix C. The 1993 translation embedded a sketch of this counterexample in five pages of the chapter on human and computer vision, but the full story takes a dozen pages of calculations. The two other new appendices discuss details of the continuous wavelet transform inversion theory in $L^{2}$ and facts about Holder and Besov spaces of functions.

Despite the new material, the same unifying ideas still bind the narrative. Meyer's original thesis was that the quantitative description of nature benefits from choosing appropriate analysis and synthesis tools and that wavelets provide a huge and versatile toolbox. There is an effective metaalgorithm, or procedure for designers of wavelet algorithms, because of the remarkable mathematical properties of wavelets. Jaffard, Meyer, and Ryan discuss these properties in greater detail than Meyer alone did. His book was a brief catalogue of the functions that had recently become available and gave some of the formulas essential to using them in particular problems.

The new version is more of an encyclopaedia. The standard of mathematical rigor is high, and the amount of detail provided in the proofs is substantial. It may be that Meyer's earlier book sought to be definitive, but the results were simply not all there. Still, the present book is not a handbook or programmer's guide, but rather an algorithm developer's guide, as not all details needed for implementations are presented. Some rather elementary results needed to implement the algorithms discussed in the book are omitted. Not mentioned at all, for example, is the symmetric extension discrete wavelet transform, which is used in both WSQ and JPEG-2000.

The standard of historical rigour is also quite high, although with so many contributors it is impossible to determine every body's role beyond dispute. Some care was evidently taken to identify the first appearance of some ideas commonly attributed to the authors, such as smooth, localised, orthonormal wavelets. "Ideas rarely have well-defined beginnings," note the authors in Chapter 12, having said the same thing in other ways in preceding chapters as well. The ideas in question concern such problems as the large-scale structure of the universe, the information content of a signal, and the nature of turbulence. Only Chapter 2 follows a timeline to show how harmonic analysis since Fourier developed into wavelet analysis. In later chapters, the historical discussion becomes increasingly idiosyncratic. With the most challenging problems, where all known methods are unsatisfactory and therefore controversial, the first published wavelet application has sometimes been neither useful nor correct. It is a pleasure to observe how gracefully and respectfully the authors describe the pioneering work in these cases.

The new collaboration shines with other examples of Meyer's familiar colloquial style.

13.1. From the Publisher.

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 generalisations 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.

Readership

Graduate students and researchers working in functional analysis and its applications, in particular to signal and image processing.

13.2. From the Preface.

This book consists of three chapters in which some seemingly unrelated scientific problems will be considered. The first chapter is devoted to image processing and more precisely to image compression and denoising. This research is motivated by the upcoming standard for still image compression. This standard will be unveiled in March 2001 and is known as JPEG-2000. In the second chapter, a few new results on the Navier-Stokes equations and other nonlinear evolution equations will be discussed and, in the third chapter, frequency modulated signals will be analysed. Motivation comes from the Virgo program of detection of gravitational waves.

How could these distinct themes possibly be studied from the same perspective? An answer is found in the contents of the three chapters.

Analysing the performances of a compression algorithm requires a model for still images. In the first chapter, our discussion will be based on the Osher-Rudin model. This model originated in a joint paper between Stanley Osher, Leonid Rudin and Emad Fatemi. It was then developed by Osher and Rudin and that is why it will be named the Osher-Rudin model. It amounts to splitting an image $f$ into a sum$f = u + v$ between two components $u$ and $v$. The first component $u$ represents the 'objects' which are contained inside the given image $f$. It is then natural to assume that $u$ belongs to the space BV of functions with bounded variation. Wavelet expansions are quite effective for analysing BV functions. On the other hand, we will prove that the 'texture + noise' component $v$ is an 'oscillating pattern'. Here 'oscillating patterns' will be defined by some Besov norm estimates. This discussion will imply that a 'wavelet thresholding' wipes away this $v$ component.

The second chapter starts with an interesting sharpening of Gagliardo- Nirenberg estimates. It continues with an improvement on Poincare's inequality. These advances rely heavily on Theorem 14 of the first chapter. The same Besov space which was used for modelling textures will be the heart of the matter here.

This specific Besov space and some related functional spaces will again be important in this second chapter. We are alluding here to a new result on the Navier-Stokes equations which was obtained by Herbert Koch and Daniel Tataru.

Feature extraction is pivotal in fluid dynamics. One would like to detect and extract the elusive 'coherent structures'. These structures are specific patterns which appear both in experimental work in fluid dynamics and in numerical simulations. In order to study coherent structures, one needs to investigate localised and oscillating solutions of the Navier-Stokes equations. This will also be done in

Einstein's general relativity implies the existence of gravitational waves. The theory predicts that such waves might be created by gigantic gravitational disasters. In the case of a collapse of a pair of binary stars, these waves would be frequency modulated signals, also named chirps. There is no doubt that detecting such signals relies on describing and analysing oscillating patterns. This important topic is discussed in Chapter 3.

The unity between our three chapters is now clear. Beyond Fourier analysis and its variants, new tools are available. These tools help in understanding and modelling oscillating patterns. Such patterns are present in many natural images. They help in explaining nonlinear evolutions. They are pivotal in chirps and frequency modulated signals.

Let me express my deep thanks to the Department of Mathematics of Rutgers University. I was proud and happy to give these three Jacqueline Lewis Lectures. Richard Falk and his colleagues have been kind and helpful.

This book is not a treatise but rather an expanded version of my informal talks. It is not a textbook. Instead it is aimed at describing a scientific vision.

Am I the sole writer of this book? This is not clear since most of the fine results which will be described or proved belong to my students or my colleagues. Moreover the 'grand unification' which is unveiled in this essay can be traced back to a program which was launched in the early forties by the 'Institute for the Unity of Science'.

Let me thank Albert Cohen and his collaborators for letting me include some still unpublished material and for his encouraging remarks. Naoki Saito read a first version of this book and suggested many improvements. The anonymous referee was kind and helpful.

I would like to dedicate my work to Jacqueline Lewis. Let me also remember Alberto Calderon. I miss so much our endless discussions about mathematics, literature and politics in Argentina.

13.3. Review by: Gilbert Walter.
Mathematical Reviews MR1852741 (2002j:43001).

This book is based on a series of lectures given by the author at Rutgers University. However, it doesn't read like a lecture series. Rather he has taken great care in converting these lectures into a clear yet fairly detailed exposition of the subject.

The book first gives a historical introduction to each of three topics which avoids jargon and detailed notation. These portions are easy to read but still full of information. A chapter is devoted to each of the topics: image compression, nonlinear partial differential equations, and frequency modulated signals. The unifying principle is that of oscillating patterns which occur in all three. In each the introductory sections are followed by more detailed sections giving the exact mathematical framework.

The first chapter on (still) image compression is the longest and most complete. It begins with an illustrative example, that of the problem of deblurring and denoising data of a super nova observed by the Hubble telescope. This is presented as a lead-in to a general problem in image and signal analysis, namely to decompose the observed image or signal $f$ into two additive components $u$ and $v$, so that $f = u + v$. Here $u$ is the "high priority", term and $v$ the "lower priority", which may include but is not restricted to additive noise. The high priority term is not necessarily the original image but may be a compressed version in which only the main features are extracted. The low priority term may be white noise but usually will have some oscillating pattern. The classical example is the extraction of an image by means of a low pass filter; in this case $v$ contains only high frequencies. But many applications require that $u$ contain high frequency elements as well; the edges are often the most important aspects of an image. It is here that wavelet analysis gives better results.

Many details of techniques suitable for image compression are given in this chapter. Sections are devoted to the Karhunen-Loève expansions and their use in the $u + v$ decomposition. These are compared to wavelet and Fourier approaches. A simple example of a discontinuous function is also given in which the sorted Fourier coefficients decay much more rapidly than wavelet coefficients (surprise). A recurring model, the Osher-Rudin model, gives a general approach to this decomposition. The $u$ in this case will belong to the space BV, while the $v$ will belong to a homogeneous Besov space.

A section on modelling textures introduces a number of Banach spaces of functions to be used subsequently and gives the inclusion relations between them, beginning and ending with homogeneous Besov spaces. The remaining sections of the first chapter deal with wavelet shrinkage, wavelet analysis, Littlewood-Paley analysis, and their relationships. Each is a readable short essay on the underlying properties which make them useful. In particular, the section on wavelet analysis could well be titled, "What every young student should know about wavelets". The final section of this chapter compares Fourier series and wavelet series and shows the latter to be more effective for analysing BV functions.

In the second chapter on nonlinear partial differential equations, the spaces and approaches of the first chapter appear again. The oscillatory pattern, which appeared in the low priority term $v$, is now taken to be the initial condition for the PDE, which is assumed to belong to the homogeneous Besov space of regularity index -1. In such cases the solutions to certain evolution equations do not exhibit blowup even though they do for other initial conditions which are smooth with compact support. These results come from new Gagliardo-Nirenberg inequalities whose proof is based on estimates involving wavelet coefficients.

The Navier-Stokes equations can be attacked by similar methods, but require a special Banach space to which the initial conditions must belong. This space consists of all generalised functions which may be written as the divergence of a vector field each of whose components belongs to the space BMO. Then for some initial condition $v$ whose norm in this Banach space is sufficiently small, which belongs to Sobolev spaces of every order, and whose divergence is 0, there is a global solution of the Navier-Stokes equations.
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