Book Description

Tracing a path from the earliest beginnings of Fourier series through to the latest research A Panorama of Harmonic Analysis discusses Fourier series of one and several variables, the Fourier transform, spherical harmonics, fractional integrals, and singular integrals on Euclidean space. The climax is a consideration of ideas from the point of view of spaces of homogeneous type, which culminates in a discussion of wavelets. This book is intended for graduate students and advanced undergraduates, and mathematicians of whatever background who want a clear and concise overview of the subject of commutative harmonic analysis. ... Read more

Reviews (1)

Nice guide for students and newcomers.
This book is not a textbook on harmonic analysis; it is an essay on how this important discipline has evolved and what are its main results and applications.

I find this book to be an excellent companion text because it includes a lot of subtle explanations that the usual textbooks lack. It is suitable also for non-mathematicians willing to get a fast introductory survey of the theory.

Its contents are: Overview of Measure Theory and Functional Analysis; Fourier Series Basics; The Fourier Transform; Multiple Fourier Series; Spherical Harmonics; Fractional Integrals, Singular Integrals, and Hardy Spaces; Modern Theories of Integral Operators; Wavelets; A Retrospective; 9 appendices.

Full of interesting comments and historical anecdotes. Extensive Bibliography.

Please take a look at the rest of my reviews. ... Read more

Book Description

An Introduction to Wavelets is the first volume in a new series, WAVELET ANALYSIS AND ITS APPLICATIONS. This is an introductory treatise on wavelet analysis, with an emphasis on spline wavelets and time-frequency analysis. Among the basic topics covered in this book are time-frequency localization, integral wavelet transforms, dyadic wavelets, frames, spline-wavelets, orthonormal wavelet bases, and wavelet packets. In addition, the author presents a unified treatment of nonorthogonal, semiorthogonal, and orthogonal wavelets. This monograph is self-contained, the only prerequisite being a basic knowledge of function theory and real analysis. It is suitable as a textbook for a beginning course on wavelet analysis and is directed toward both mathematicians and engineers who wish to learn about the subject. Specialists may use this volume as a valuable supplementary reading to the vast literature that has already emerged in this field.

Key Features
* This is an introductory treatise on wavelet analysis, with an emphasis on spline-wavelets and time-frequency analysis
* This monograph is self-contained, the only prerequisite being a basic knowledge of function theory and real analysis
* Suitable as a textbook for a beginning course on wavelet analysis ... Read more

Reviews (2)

Infinite in all directions.
I have used the book, teaching a beginning course in wavelets, and it went over well with the students: they like that the proofs are clearly spelled out, and that the presentation is systematic. I supplemented it with S Mallat and Y Meyer's books. That was to give the course a more algorithmic slant. However, I feel that there are many valid approaches, and that the subject is infinite in all directions.

Great introduction for Wavelet
For those who would like to learn the beauty of Wavelet theory, which is a great finding in Mathematics, this book would shed some light on it. Wavelet is elegant, a simple formula leads to so much interests, theapplication of Wavelet is everywhere, particularly in modern communicationsand signal processing, and there is still more waiting for people toexplore. Enjoy the new journey. ... Read more

Book Description

Time Warps, String Edits and Macromolecules is a young classic in computational science. The computational perspective is that of sequence processing, in particular the problem of recognizing related sequences. The book is the first, and still best compilation of papers explaining how to measure distance between sequences, and how to compute that measure effectively. This is called string distance, Levenshtein distance, or edit distance. The book contains lucid explanations of the basic techniques; well-annotated examples of applications; mathematical analysis of its computational (algorithmic) complexity; and extensive discussion of the variants needed for weighted measures, timed sequences (songs), applications to continuous data, comparison of multiple sequences and extensions to tree-structures. This theory finds applications in molecular biology, speech recognition, analysis of bird song and error correcting in computer software. ... Read more

Reviews (1)

Still a useful source of information
This book, originally published in 1983, was reissued in 1999, no doubt because of the importance of genetic sequencing in recent years. What is neat about the book is it shows how algorithms from one field can be applied to solve problems in another, possibly totally disparate field, one example being computational linguistics and sequence algorithms in computational biology.

A general overview of sequence comparison is given in chapter 1 with applications to molecular biology, human speech, computer science, coding theory, gas chromotography, and bird songs discussed. The author discusses how deletion-insertion, compression-expansion, and substitution are employed in sequence comparison. Different metrics are introduced, such as the Levenshtein distance. Dynamic programming, which pretty much dominates the book, is introduced here also.

Part 1 of the book discusses sequence comparison in molecular biology. The use of dynamic programming is emphasized and its importance continues to this day. The advantages of using the dynamic programming method are outlined, and it is shown how to find the substring in a longer sequence with most optimum agreement to a shorter sequence. In addition, given an RNA molecule with a known nucleotide sequence, methods are discussed for predicting the way different parts of the molecule will bond to each other. These methods are based on dynamic programming. Mathematicians considering doing research on or about entering the field will profit from the section on the biological background. The treatment of RNA secondary structures is excellent.

In part 2, the emphasis is on speech processing and what is called "time-warping", which is a technique for comparing functions by altering the time axis. An interesting application is given to the comparison of bird songs. An algorithm is given for adjusting the time scales for two songs to arrange them in the most optimal alignment. In addition, the differences between compression and expansion and deletion and insertion are discussed in this part.

In part 3, a modified Smith-Waterman algorithm is employed to find similar portions in two sequences. Called local alignment in computational biology, it is shown in detail how to define the recurrences for the alignment and how to keep track of the pointers for backtracking. This part also generalizes the operations of substitution and Levenshtein distance. In addition, the strategy of doing sequence comparison by allowing transpositions is discussed. Such a strategy entails a generalized concept of trace, wherein trace lines can intersect each other, leading to entangling of the traces into knots or plaids. The usual dynamic programming techniques must then be extended to deal with these complications. One particular algorithm for this is discussed, called CELLAR, which involves the construction of a directed graph whose paths correspond to admissible sequences of generalizations of traces, called cuts. The computational complexity of this algorithm is discussed. In addition, an O(n^2/logn) algorithm is given for computing string-edit distances.

The last part of the book deals with studying comparisons between random sequences. Combinatorial arguments are used to derive upper bounds on the expected length of the longest common subsequences of two random sequences. Other miscellaneous results dealing with comparing common subsequences of two random sequences are given. ... Read more

Book Description

Abstract theory remains an indispensable foundation for the study of concrete cases. It shows what the general picture should look like and provides results that are useful again and again. Despite this, however, there are few, if any introductory texts that present a unified picture of the general abstract theory.A Course in Abstract Harmonic Analysis offers a concise, readable introduction to Fourier analysis on groups and unitary representation theory. After a brief review of the relevant parts of Banach algebra theory and spectral theory, the book proceeds to the basic facts about locally compact groups, Haar measure, and unitary representations, including the Gelfand-Raikov existence theorem. The author devotes two chapters to analysis on Abelian groups and compact groups, then explores induced representations, featuring the imprimitivity theorem and its applications. The book concludes with an informal discussion of some further aspects of the representation theory of non-compact, non-Abelian groups. ... Read more

Book Description

The past decade has witnessed the rpid development of a new mathematical tool, called wavlet analysis, for analyzing complex signals. It has begin to play a serious role in applications ranging from communications to geophysics, and from simulations to image processing. Like Fourier analysis (of which it is a generalization), or musical notation, wavelet analysis provides a method for representing a set of complex phenomena in a simpler, more compact, and thus more efficient manner. This text introduces the ideas and methods of wavelet analysis, relates them to previously known methods in mathematics and engineering, and shows how to apply wavelet analysis to digital signal processing. It begins by describing the multiscale (sometimes called "fractal") nature of information in many aspects of thereal world; it then turns to the algebra and analysis of wavelet matrices, scaling and wavelet functions, and the corresponding analysis of square-integrable functins on a space. The discussion then turns from the continuous to the discrete and shows how a properly selected set of wavelets can be used to represent -- and even differentiate -- a wide range of signls efficiently and effectively. The last part of the book presents a wide variety of applications of wavelets to probllems in data compression and telecommunications. ... Read more

Reviews (5)

One of the best!
The Wells-Resnikoff book on wavelets is in my opinion
one of the best:-- both with regard to the math--the proofs and all that!--, and the applications. The ideas in the book are timeless, and, I expect, it will be a great book for decades to come. It is very strong on both scores. And it is beautifully written: It is all there,- the history, the
motivation, the philosophical points, the applications, the guided tours with friendly author-explainations of what it is all about. For me, this book was a page-turner. I read it on
a long plane flight,- on my way to a wavelet conference in Guangzhou, China. -- And I didn't have jet-lag when I arrived
more than a day later. It may be a bit demanding on some students, but it will be well worth their effort!

Not good.
If the authors were less enthusiastic about mathematics for its own sake, they probably would have written a better book. This one is said to be suitable for undergraduates, but I don't think so. I have a master's degree in electrical engineering, and several of the mathematical concepts they mention were just vaguely familiar names to me. That would be OK if I were persuaded that those concepts were essential to an understanding of wavelets, but I'm not. The book seems somewhat ill organized as well; new terminology should be defined before it's used, not fifty pages later.

Strong mathematical foundations
This is a wonderful book for the pure mathematicians out there. Unlike many other texts, this book spans the entire range of explanation from high-level outline through the algebraic foundations to real-world applications. If you're just interested in applying wavelets with a minimum of fuss, you may want another book. But if you want to really understand the theory behind them, this is the one for you.

beyond signal processing
In most books on wavelets, signal and image processing are the only topics of application considered, because they are written by people from that field. However, the applicability of wavelets is much wider. For instance, the solution of partial differential equations, which has allowed the entrance of wavelets to the finite element world. This and other links to classical math problems, such as fractal geometry, are carefully addressed in this book besides the conventional signal processing issues. I recommend it for people of various fields.

The Best Introductory Book on the Subject
This is The Read-First book on Wavelets! ... Read more

Book Description

The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations, and to motivate the study of harmonic analysis on more general spaces having an analogous structure, e.g., symmetric spaces. ... Read more

Reviews (1)

A superb classic in harmonic analysis.
This book deals with the extension of real and complex methods in harmonic analysis to the many-dimensional case. So, its pre-requisites are a strong background in real and complex analysis and some acquaintance with elementary harmonic analysis, that is, this book is intended for graduate students and working mathematicians. Maybe some advanced undergraduates could cover certain parts of the material.

This book is one component of the Stein trilogy on harmonic analysis (together with "Singular Integrals and Differentiability Properties of Functions" and "Harmonic Analysis", both also reviewed by myself), and as such it must be regarded as an authoritative reference on the subject since Elias Stein and Guido Weiss are two of the leading experts in the field, and the material they selected was taken from their teaching and research experience.

The contents of the book are: The Fourier Transform; Boundary Values of Harmonic Functions; The Theory of H^p Spaces on Tubes; Symmetry Properties of the Fourier Transform; Interpolation of Operators; Singular Integrals and Systems of Conjugate Harmonic Functions; Multiple Fourier Series.

Includes motivation and full explanations for each topic, excercises for each chapter, called "further results", and extensive references. Outstanding printing quality and nice clothbound.

These three volumes should be present in every analyst's library.

Please take a look to the rest of my reviews (just click on my name above). ... Read more

Book Description

This book is a primer in harmonic analysis using an elementary approach. Its first aim is to provide an introduction to Fourier analysis, leading up to the Poisson Summation Formula. Secondly, it makes the reader aware of the fact that both, the Fourier series and the Fourier transform, are special cases of a more general theory arising in the context of locally compact abelian groups. The third goal of this book is to introduce the reader to the techniques used in harmonic analysis of noncommutative groups. There are two new chapters in this new edition. One on distributions will complete the set of real variable methods introduced in the first part. The other on the Heisenberg Group provides an example of a group that is neither compact nor abelian, yet is simple enough to easily deduce the Plancherel Theorem.

Book Description

This intriguing, accessible work leads readers to an excellent grasp of the fundamental notions of infinity used in the calculus and in virtually all other mathematical disciplines. Designed to make minimal demands on readers’ mathematical abilities, this perceptive volume employs an intuitive and descriptive approach that will appeal to students, teachers, and the mathematically inclined. Each chapter’s teachings are supplemented by challenging problems, with solutions at the end of the book. 83 text figures. 1962 ed.
... Read more

Reviews (2)

An examination of the role of infinity in mathematics.
Zippin's engaging text examines how infinity arises in mathematics. It requires only a solid grounding in high school mathematics and a willingness to think. However, it will most profitably be read by those who are familiar with calculus.

After a brief overview, Zippin begins within the natural numbers. He shows how inductive reasoning is used to search for patterns that can be used to prove results about infinite sequences and series. He examines limits of sequences and series from a geometric point of view. In particular, he considers geometric series and the Fibonacci sequence, using the latter to explore the properties of the golden rectangle. He concludes the text by discussing recursive definition, proofs by mathematical induction, and the pigeon hole principle.

I found the material intriguing and the exposition generally clear. However, there were places where I felt that definitions were imprecise. His proof that the square root of 2 is irrational, while elegant, is harder to generalize than other proofs of that result. Zippin uses numerous examples to illustrate the results that he proves.

Zippin's decision to examine limits from a geometric standpoint provides an interesting alternative to the analytical approach taken in calculus courses. It also helps the reader understand his arguments in his chapter on how the golden rectangle is related to the golden mean and the Fibonacci numbers.

The exericses, for which answers are provided in the back of the text, are thought-provoking and some are quite challenging. I found reading his solutions instructive.

Zippin provides a now dated (the text was published in 1962) bibliography so that the reader can explore the topics he discusses further. The reader may wish to consult the texts Invitation to Number Theory by Oystein Ore and Numbers: Rational and Irrational by Ivan Niven while reading this text.

GREAT INTRODUCTION TO INFINITY!
This book was my first introduction to the infinite. Although it is easy enough for undergraduate study, all will find Zippin's book scintillating and fascinating. A great read for anyone interested in this subject! ... Read more

Book Description

An indispensable guide to understanding wavelets
Elements of Wavelets for Engineers and Scientists is a guide to wavelets for "the rest of us"-practicing engineers and scientists, nonmathematicians who want to understand and apply such tools as fast Fourier and wavelet transforms. It is carefully designed to help professionals in nonmathematical fields comprehend this very mathematically sophisticated topic and be prepared for further study on a more mathematically rigorous level.
Detailed discussions, worked-out examples, drawings, and drill problems provide step-by-step guidance on fundamental concepts such as vector spaces, metric, norm, inner product, basis, dimension, biorthogonality, and matrices.
Chapters explore . . .
* Functions and transforms
* The sampling theorem
* Multirate processing
* The fast Fourier transform
* The wavelet transform
* QMF filters
* Practical wavelets and filters
. . . as well as many new wavelet applications-image compression, turbulence, and pattern recognition, for instance-that have resulted from recent synergies in fields such as quantum physics and seismic geology.
Elements of Wavelets for Engineers and Scientists is a must for every practicing engineer, scientist, computer programmer, and student needing a practical, top-to-bottom grasp of wavelets. ... Read more

Reviews (2)

Great Book
Wavelets is a mathematically intensive subject. As such many books on the subject present the topic as an endless list of equations with very little insight.

This book on the other hand presents the material in a fashion that is digestable to the average engineer. This is a credit to their writting and teaching skill. They write as if their first goal is to let you understand (teach) not simply write.

I would say that the first books to read on this subject would be this one and the book by Walker. These books are sufficient to get a good understanding of Wavelets and to be able to use them. They also pave the way for more advanced books on the subject if desired.

As good as it gets
This is an excellent short introductory book for anyone who wants to understand and use wavelets via the filtering route. The presentation is entirely self-contained, and includes completely worked out wavelet examples and answers to all exercises. The authors develop wavelets as multirate quadrature mirror filters (explained within the text). Sample matlab source code is provided. Like Mix's text on Random Signal Processing, this one shines as being practical, easy-to-understand, filled with examples and prepares the reader for more advanced work. ... Read more

Book Description

In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Rucker acquaints us with Gödel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of infinity, he maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations.

Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Gödel's incompleteness theorems. His personal encounters with Gödel the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism. ... Read more

Reviews (14)

A passionate introduction to the theme of infinity
The book mentiones : Infinity commenly inspires feelings of awe, futility and fear. Reading of the book makes one agree to it. The book is written for a reader who is philosophically curious and patient in reading. After introducting the various context ( spatial, temporal , physical) where one encounter the issue of infinity, the author explain clearly the debate of potential vs actual infinity. Here author points out about the Greek philosophical tendencies. Chapter two discusses the revolution brought by Cantor's works. He explains the concept using a lot of symbols, diagrams and illustrations. The reader is made to understand the notion of transfinite number. The chapter ends with an extract from his novel White Light which deals with the idea of the chapter. Next chapter discusses the kind of paradoxes one encounter in thinking the theme of infinity within modern mathematical logical framework. Chapter four discusses the implications of Godel's theorems in question of Robot consciousness. He gives details about his personal interactions with Godel. He mentiones about his dream about Godel the day before Godel's death. This is most humanistic chapter. Last chapter discusses the abstract philosophical reflections. There are two well written excursion chapters : one on Cantor's set theory and one on Godel's Incompleteness theorems. Every chapter has well thought puzzles and paradoxes section.

Infinity made simple and understandable
In many ways, infinity is the most abstract concept of all. Many of the advances in understanding how to manipulate infinities had unpleasant consequences. As the legend goes, the first one to announce that there are infinite non-repeating decimals was rewarded by being drowned. Georg Cantor, the first to prove that there are different levels of infinity, faced extreme criticism and ultimately went mad. Fortunately, Rudy Rucker provides a gentle introduction to this concept, one that can be read by most with the only consequence being enlightenment.
The entire range of infinities (what a phrase!) is covered in this book. From the simplest infinity (omega), to the multi-universe theories of quantum theory. All are put forward in a very readable style, although there are times when one must slow down and read very carefully if one is to understand. Rucker's encounters with Kurt Godel is a welcome contrast with the common depiction that he was a dry, humorless man. It is refreshing to hear that he laughed and had a sense of humor.
Many different test scenarios have been put forward to determine if a computer is indeed intelligent. At this time, I would propose that any machine that can understand the concept of infinity must be considered intelligent. Any human wishing to pass that test need only read this book. It should be required reading in all undergraduate mathematics programs.

Published in Journal of Recreational Mathematics, reprinted with permission.

At the intersection of parallel lines...
Rudy Rucker, son of a cleric and mathematics whiz kid, produced this book on 'Infinity and the Mind' years ago, but reading and re-reading it, I continue to get insights and the chance to wrap my mind around strange concepts.

'This book discusses every kind of infinity: potential and actual, mathematical and physical, theological and mundane. Talking about infinity leads to many fascinating paradoxes. By closely examining these paradoxes we learn a great deal about the human mind, its powers, and its limitations.'

This book was intended to be accessible by those without graduate-level education in mathematics (i.e., most of us) while still being of interest to those even at the highest levels of mathematical expertise.

Even if the goal of infinity is never reached, there is value in the journey. Rucker provides a short overview of the history of 'infinity' thinking; how one thinks about divinity is closely related often, and how one thinks about mathematical and cosmological to-the-point-of-absurdities comes into play here. Quite often infinite thinking becomes circular thinking: Aquinas's Aristotelian thinking demonstrates the circularity in asking if an infinitely powerful God can make an infinitely powerful thing; can he make an unmade thing? (Of course, we must ask the grammatical and logical questions here--does this even make sense?)

Rucker explores physical infinities, spatial infinities, numerical infinities, and more. There are infinites of the large (the universe, and beyond?), infinities of the small (what is the smallest number you can think of, then take half, then take half, then take half...), infinities that are nonetheless limited (the number of divisions of a single glass of water can be infinite, yet never exceed the volume of water in the glass), and finally the Absolute.

'In terms of rational thoughts, the Absolute is unthinkable. There is no non-circular way to reach it from below. Any real knowledge of the Absolute must be mystical, if indeed such a thing as mystical knowledge is possible.'

At the end of each chapter, Rucker provides puzzles and paradoxes to tantalise and confuse.

* Consider a very durable ceiling lamp that has an on-off pull string. Say the string is to be pulled at noon every day, for the rest of time. If the lamp starts out off, will it be on or off after an infinite number of days have passed?

Rucker explores the philosophical points of infinity with wit and care. He explores the ideas behind and implications of Gödel's Incompleteness Theorem, and leads discussion and excursion into self-referential problems and set theory problems and solutions.

He also discusses, contrary to conventional wisdom, the non-mechanisability of mathematics. We tend to think in our day that mathematics is the one mechanical-prone discipline, unlike poetry or creative arts and more 'human' endeavours. But Rucker discusses the problems of situations which require decision-making and discernment in mathematical choices that no machine can (yet!) make.

* Consider the sentence S: This sentence can never be proved. Show that if S is meaningful, then S is not provable, and that therefore you can see that S must be true. But this constitutes a proof of S. How can the paradox be resolved?

This is a beautifully complex and intriguing book on the edges of mathematics and philosophical thinking, which is nonetheless accessible and intellectually inviting. You'll wonder why math class was never this fun!

a mind-blowing trip to the infinite
What is infinity? How do we train our minds to understand the idea? This one of the hardest questions to answer for non-professional mathematicians, and one that Rucker address superbly - and, believe it or not entertainingly in this excellent book. And once you think you grasped that, how about a higher level infinity? Next one? Infinite series of higher level infinities? Sound very scary, and it is. It takes an amazing capacity to explain these concepts to a (relative) layman, and Rucker has it in abundance. An exhilarating intellectual tour de force, perhaps comparable to climbing mount Everest - infinite number of times, with deep philosophical, and perhaps, religious connections, presented in a light, funny, and yet rigorous manner. The book also provides a history of the concept of the infinite, and interesting people who developed it. A must read for a curious mind.

A perfect book for someone like me
I know very little about any of the subjects discussed in this book, although I do have a degree in philosophy of science, and I liked this book a lot.

I can't believe I made it through 7 years of senior school and 2 years of degree level maths and nobody ever bothered to tell me about infinity, transfinite numbers, set theory and its relationships with, and underpinning of other branches of mathematics in a way I could understand rather than simply regurgitate. Rucker on the other hand manages to do this in 362 pages.

I slso found the stuff about Godel and the impossibility of complete formulisms very useful, not only philosophically, but also just for my own peace of mind. ... Read more

Book Description

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 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.

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. ... Read more

Reviews (3)

Lots of new additions.
Of the many recent books on wavelets, this one by Jaffard, Meyer et al stands out. This is a new edition of earlier ones, starting with a French one, then a first translation, and now this expanded version. There is lots of new material, covering new developments, both with regards to theory and applications. The authors communicate the main points in a clear and attractive way, and it stresses lucid presentation of ideas over formulas. And with an infectious enthusiasm! The presentation is especially refreshing, in that the authors do not hide behind excessive technicalities and mathematical jargon. It is the central idea that shines through. And yet the mathematically sophisticated reader is paid his dues,-- in some cases via the appendices. At that point, the technical mathematical points have been well motivated. The emphasis of the book is the mathematics behind the many algorithms that emerged from the wavelet trends since the 1980ties. And lots of new stuff added (for example JPEG-2000), right up to the present. The book is rich in applications, many of which are not found elsewhere. The inclusion of motivation and historical remarks is welcome, and makes the book enjoyable reading.

Jaffard et al
Of the many recent books on wavelets, this one
by Jaffard et al stands out. It communicates the
main points in a clear and attractive way, and it stresses lucid presentation of
ideas over formulas. The authors do not hide behind technicalities
and mathematical jargon. And yet the mathematically sophisticated
reader is paid his dues,-- in some cases via the appendices. At that point, the technical mathematical points have been well motivated. The emphasis of the book is the mathematics behind
the many algorithms that emerged from the wavelet trends
since the 1980ties.
The book is rich in applications, many of which
are not found elsewhere. The inclusion of motivation
and historical remarks is also welcome.

Good covering of wavelets applications
Although this book emphasis is on algorithms and applications, the general presentation of wavelets for signal processing is great. The introduction to the subject is really excellent and great care has been given to explaining the use of wavelets for applications in signal processing. On the other side i found the last chapters dealing with applications a little too light for being useful...

Anyway, a great book for "wavelets and applications in signal processing"...(also go to "Time-Frequency Time-scale analysis" by Patrick Flandrin which is wonderful). ... Read more

Book Description

Professor Zygmund's Trigonometric Series, first published in Warsaw in 1935, established itself as a classic. It presented a concise account of the main results then known, but on a scale that limited the amount of detailed discussion possible. A greatly enlarged second edition (Cambridge, 1959) published in two volumes took full account of developments in trigonometric series, Fourier series, and related branches of pure mathematics since the publication of the original edition. These two volumes, bound together with a foreword from Robert Fefferman, outline the significance of this text. Volume I, containing the completely re-written material of the original work, deals with trigonometric series and Fourier series. Volume II provides much material previously unpublished in book form. ... Read more

Book Description

This is the first comprehensive exposition of the application of spherical harmonics to prove geometric results.The author presents all the necessary tools from classical theory of spherical harmonics with full proofs.Groemer uses these tools to prove geometric inequalities, uniqueness results for projections and intersection by planes or half-spaces, stability results, and characterizations of convex bodies of a particular type, such as rotors in convex polytopes. Results arising from these analytical techniques have proved useful in many applications, particularly those related to stereology. To make the treatment as self-contained as possible the book begins with background material in analysis and the geometry of convex sets. ... Read more

Book Description

An ideal refresher or introduction to contemporary Fourier Analysis, this book starts from the beginning and assumes no specific background. Readers gain a solid foundation in basic concepts and rigorous mathematics through detailed, user-friendly explanations and worked-out examples, acquire deeper understanding by working through a variety of exercises, and broaden their applied perspective by reading about recent developments and advances in the subject. Features over 550 exercises with hints (ranging from simple calculations to challenging problems), illustrations, and a detailed proof of the Carleson-Hunt theorem on almost everywhere convergence of Fourier series and integrals of L<^>p functions—one of the most difficult and celebrated theorems in Fourier Analysis. A complete Appendix contains a variety of miscellaneous formulae.L<^>p Spaces and Interpolation. Maximal Functions, Fourier transforms, and Distributions. Fourier Analysis on the Torus. Singular Integrals of Convolution Type. Littlewood-Paley Theory and Multipliers. Smoothness and Function Spaces. BMO and Carleson Measures. Singular Integrals of Nonconvolution Type. Weighted Inequalities. Boundedness and Convergence of Fourier Integrals.For mathematicians interested in harmonic analysis. ... Read more

Book Description

Conceived by the author as an introduction to "why the calculus works" (otherwise known as "analysis"), this volume represents a critical reexamination of the infinite processes encountered in elementary mathematics. Part I presents a broad description of the coming parts, and Part II offers a detailed examination of the infinite processes arising in the realm of number--rational and irrational numbers and their representation as infinite decimals. Most of the text is devoted to analysis of specific examples. Part III explores the extent to which the familiar geometric notions of length, area, and volume depend on infinite processes. Part IV defines the evolution of the concept of functions by examining the most familiar examples--polynomial, rational, exponential, and trigonometric functions. Exercises form an integral part of the text, and the author has provided numerous opportunities for students to reinforce their newly acquired skills. Unabridged republication of Infinite Processes as published by Springer-Verlag, New York, 1982. Preface. Advice to the Reader. Index.
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Reviews (1)

Good, but not great
Most of the books I have given a 4-star rating to would be in the category of "If they'd only done this thing differently, I'd have given it 5." This book is different. I'm taking literally the definition "5 = I love it!" and on that basis this book cannot get 5 stars, though I can't say exactly _what_ I don't quite like.

The book does a good job of taking the reader through the steps needed to explain "why the calculus works" -- and it really doesn't require calculus to understand most of this book, though you need calculus to understand the _motivation_ for a lot of what the book does. You need only have a decent grounding in high school geometry and algebra to understand the book. Still, it isn't an _easy_ read. I've found myself putting the book down and coming back to it later several times. Yet I can't say that the book is hard to read because it's badly written; I think it's written decently well. I think it's just that the topics it covers are harder going than a lot of the math I read.

So, while I don't _love_ this book, I'd recommend it to someone who wants to know something about the motivation for all those weird epsilons and deltas and the non-intuitive definitions of topics like continuity. It does that job, and does it as well as any book I know of. ... Read more

Book Description

Fourier Integrals in Classical Analysis is an advanced treatment of central problems in harmonic analysis. The main theme of the book is the interplay between ideas used to study the propagation of singularities for the wave equation and their counterparts in classical analysis. Using microlocal analysis, the author in particular studies problems involving maximal functions and Riesz means using the so-called half-wave operator. This self-contained book starts with a rapid review of important topics in Fourier analysis. The author then presents the necessary tools from microlocal analysis, and goes on to give a proof of the sharp Weyl formula which he then modifies to give sharp estimates for the size of eigenfunctions on compact manifolds. Finally, the tools that have been developed are used to study the regularity properties of Fourier integral operators, culminating in the proof of local smoothing estimates and their applications to singular maximal theorems in two and more dimensions. ... Read more

Book Description

An Introduction to Non-Harmonic Fourier Series, Revised Edition is an update of a widely known and highly respected classic textbook.

Throughout the book, material has also been added on recent developments, including stability theory, the frame radius, and applications to signal analysis and the control of partial differential equations. ... Read more