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1. Schaum's Outline of Fourier Analysis
2. Fourier Analysis
3. An Introduction to Fourier Analysis
4. Fourier Analysis and Its Applications
5. Fourier Analysis (Graduate Studies
6. Exercises in Fourier Analysis
7. A First Course in Fourier Analysis
8. Modern Fourier Analysis (Graduate
9. Fourier Analysis and Imaging
10. Classical Fourier Analysis (Graduate
11. Introduction to Fourier Analysis
12. Fourier Analysis: An Introduction
13. Introduction to Fourier Analysis
14. Fourier Analysis, Self-Adjointness
15. Classical and Modern Fourier Analysis
16. Who Is Fourier?: A Mathematical
17. Fourier Analysis on Finite Groups
18. Fourier Analysis on Number Fields
19. A First Course in Wavelets with
20. Fourier Analysis

1. Schaum's Outline of Fourier Analysis with Applications to Boundary Value Problems
by Murray Spiegel
Paperback: 208 Pages (1974-03-01)
list price: US$19.95 -- used & new: US$10.53
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Asin: 0070602190
Average Customer Review: 4.0 out of 5 stars
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Ace your course in Fourier analysis with this powerful study guide! With its clear explanations, hundreds of fully solved problems, and comprehensive coverage of the applications of Fourier series, this useful tool can sharpen your problem-solving skills, improve your comprehension, and reduce the time you need to spend studying. It also includes hundreds of additional practice problems for you to work on your own, at your own speed to help you get ready for tests. Featuring theorem proofs as well as real-world application examples, this comprehensive guide is also the perfect tutor for brushing up for graduate or professional exams! ... Read more

Customer Reviews (4)

3-0 out of 5 stars Several exercises
This book have a several exercises.
IF you prepare a test, i recommend you purchase this book

5-0 out of 5 stars Good for Fourier analysis and much more
This Schaum's outline is unique in that you not only get a thorough coverage of Fourier analysis, but of other orthogonal functions such as Bessel, Legendre, Hermite, and Laguerre. The first chapter would be interesting to students of partial differential equations because of its excellent treatment of boundary value problems and of the different types of partial differential equations. It makes an interesting first taste of PDE solution methods. Chapter two is a traditional treatment of Fourier series and its applications. As in the first chapter, it is the applications that make the chapter unique as the Fourier series is used to solve problems in heat flow, Laplace's equations, and vibrating systems. Chapter three has a good discussion of why you would actually care if a function is orthogonal. Chapter four discusses special functions and how they are evaluated. Chapters five and six are all about applications of the Fourier integral and of the Bessel function respectively. Chapter seven uses the Legendre functions to solve problems such as finding the potential interior and exterior to a sphere given a specific charge distribution. Chapter eight finishes the guide with a discussion of Hermite and Laguerre polynomials, more from a properties standpoint than from an applications standpoint. The reader of this guide should already be knowledgeable of Calculus and differential equations, and should probably have some kind of background in physics or engineering to get the most from the book. It would be a good supplement for the student of partial differential equations or signal processing as well as the student of Fourier analysis. I think what really sets this book apart is its ability to act as a stand-alone guide to the student with the required prerequisites and to actually to inspire you to study applied mathematics more. As for my own story, I picked up a previous edition of this book 17 years ago for a coworker that thought it might be helpful in a class he was taking. He decided he didn't want it and so I began thumbing through it. I found the applications sections to be so interesting and inspiring that I wound up going back to graduate school and eventually picked up three master's degrees! If you like this Schaum's outline, you might also want to pick up "Schaum's Outline of Advanced Mathematics for Engineers and Scientists" by the same author. It is also full of mathematics inspired by real world problems in need of solution.

4-0 out of 5 stars Step-by-step explanations
This text is a good supplement to understanding the use of Fourier analysis and how it is used in real-world applications.The explanations are to the point and the solved problems are all fairly easy to follow.

At the end of the chapter, there are exercises to test your knowledge, and most of the answers are in the back of the book.Modeling the exercises on the problems, you can usually work out what you should do for the exercise.

This is a good study guide.

4-0 out of 5 stars Good study aid!
Very helpful to me in my medical imaging and signal processing assignments! A good buy for any student of engineering or science, particuarly useful to the study of signal analysis. ... Read more

2. Fourier Analysis
by T. W. Körner
Paperback: 591 Pages (1989-11-24)
list price: US$80.00 -- used & new: US$32.77
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Asin: 0521389917
Average Customer Review: 4.0 out of 5 stars
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The author has provided a shop window for some of the ideas, techniques and elegant results of Fourier analysis, and for their applications. These range from number theory, numerical analysis, control theory and statistics, to earth science, astronomy, and electrical engineering.Each application is placed in perspective with a short essay. The prerequisites are few (the reader with knowledge of second or third year undergraduate mathematics should have no difficulty following the text), and the style is lively and entertaining. ... Read more

Customer Reviews (7)

5-0 out of 5 stars Possibly the most fun you can have reading a math book
The author weaves together stories, application, and mathematics to give a fairly complete vision of Fourier analysis.All the mathematics is done without requiring measure theory, and the motivation is always in the forefront.My personal favorite parts are the on the building of the transatlantic cable and the example of outstanding statististical analysis.Make sure to purchase the exercises book, even if it is only for the jokes.

1-0 out of 5 stars Disappointing
This book is definitely not for those who really want to learn Fourier analysis. As soon as you read the preface you start thinking it is going to be a bad book, as the author states it explcitily "this is not a book for this, not a book for that..." It sticks to a list of theorems without proof many times or with ugly proofs other times. It is not structured and not even explain the motivation behind each result.

If someone really wants to know Fourier anlisys I would recommend "Fourier and Wavelet Analysis" from Bachman, Narici and Beckstein

4-0 out of 5 stars A beautiful panorama, but unhelpful in some respects
This book is valuable for its emphasis on interesting applications. The treatment of the mathematical basics of Fourier analysis is too hasty to be of much value as a first text, but this is only to leave room for the many beautiful applications. To set the tone, Weyl's equidistribution theorem appears on page 11 (sic). Later we see the classical problems of 19th century physics, but also little samples of Brownian motion, Monte Carlo methods, cryptography and other modern things. When Korner feels like it, he includes historical remarks and anecdotes that are pleasantly told. I wish this cosy atmosphere could also have been upheld in the mathematical details. Unfortunately, when it comes to the proofs, Korner seems to have a bit of a macho attitude towards long calculations. A few words of explanation here and there would probably be of great help when one is lost in page after page of technical lemmas consisting of nothing but formulas and curious integral approximations.

5-0 out of 5 stars Excellent!
This book makes great reading. There is a fair amount of (well written) high level mathematics, but also a number of sections of a more historical or narrative nature, and a wonderful sense of humor pervades the work. Theaccount of the laying of the transatlantic cable in the nineteenth centuryand the technical problems associated with it is priceless. Severalsections are devoted to the life of Fourier. There is also a companionvolume entitled ``Exercises for Fourier analysis''.

4-0 out of 5 stars Disappointment.
I'm an undergraduate student for electrical engineering in Tel Aviv University. I find this book very interesting and fun to read. However, I must say, that it has a serious lack of examples, and there aren't anyexercises. In bottom line - it can be an excellent book for professionals,as a student ? it's almost impossible to study from it without the lecturenotes. ... Read more

3. An Introduction to Fourier Analysis and Generalised Functions (Cambridge Monographs on Mechanics)
by M. J. Lighthill
Paperback: 80 Pages (1958-01-01)
list price: US$24.99 -- used & new: US$19.35
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Asin: 0521091284
Average Customer Review: 5.0 out of 5 stars
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This monograph on generalised functions, Fourier integrals and Fourier series is intended for readers who, while accepting that a theory where each point is proved is better than one based on conjecture, nevertheless seek a treatment as elementary and free from complications as possible. Little detailed knowledge of particular mathematical techniques is required; the book is suitable for advanced university students, and can be used as the basis of a short undergraduate lecture course. A valuable and original feature of the book is the use of generalised-function theory to derive a simple, widely applicable method of obtaining asymptotic expressions for Fourier transforms and Fourier coefficients. ... Read more

Customer Reviews (1)

5-0 out of 5 stars Great for self study
Nice little book. Concise and clear. In the typical English style, many of the proofs are left to the reader. Lighhill will assume what he finds obvious is also obvious to the reader. This was not always the case for me. He does this alot in the proving theorems so have a pencil ready. It is alot fun to work through the proofs becuase he gives rough sketches and leaves you to fill in the details. He manages this without loss of clarity or rigor. It is good pedagogy for anyone familiar with basic analysis of real variables and some knowledge of complex variables. ... Read more

4. Fourier Analysis and Its Applications (Pure and Applied Undergraduate Texts)
by Gerald B. Folland
Hardcover: 433 Pages (2009-01-13)
list price: US$72.00 -- used & new: US$65.00
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Asin: 0821847902
Average Customer Review: 5.0 out of 5 stars
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This book presents the theory and applications of Fourier series and integrals, eigenfunction expansions, and related topics, on a level suitable for advanced undergraduates. It includes material on Bessel functions, orthogonal polynomials, and Laplace transforms, and it concludes with chapters on generalized functions and Green's functions for ordinary and partial differential equations. The book deals almost exclusively with aspects of these subjects that are useful in physics and engineering, and includes a wide variety of applications. On the theoretical side, it uses ideas from modern analysis to develop the concepts and reasoning behind the techniques without getting bogged down in the technicalities of rigorous proofs. ... Read more

Customer Reviews (3)

5-0 out of 5 stars Folland is The One
Thank God for this book.Folland is the grand-master of my life.I'm a grad student in applied math, studying for qualifying exams, and this book has been extremely helpful to me because it is perfectly rigorous and also crystal clear.When I say Folland is The One, I am referring to The Matrix.This book is that good.His treatment of distributions (otherwise known as "generalized functions") is especially nice.

5-0 out of 5 stars one of the best books on the subject
I've been asked to teach a course on Fourier analysis,I knew nothing on the subject so I took about 15 books on this subject and went over all of them. My conclusion was that Folland's book is the best!. He explain the theory yet never forget for a minute the intuitive side of the subject. The book contains almost all the important issues and notions of the subject. If you have a solid background in vector calculus and you know some basic facts about ODE this is a very good book to learn the subject from. Moreover the book give the reader some of the important motivations to the basic ideas of functional analysis such as generatingfunctions distrbutions it gives the connection alsobetween linear algebra and the basic ideas that lies at the foundations for understanding normed function spaces and more. Moreover the book draw the line , in a very elegant way, between functional analysis PDE and Fourier analysis. Main subjects are:Fourier series,orthogonal sets Fourier and Laplace transforms,convolution, generating functions,Green functions, and more. Very recommended!.

5-0 out of 5 stars Simple yet Instructive and exhaustive
Concepts are not hidden under obscure mathematical notation: they are stated explicitly in plain english and illustrated with examples. I read a couple of other books on this topic (and PDE) without really understanding the subtleties. With this book everything becomes magically clear and obvious -as you read- and don't feel like you need take another course in real analysis to understand this topic. Bonus: you get solutions to exercices. ... Read more

5. Fourier Analysis (Graduate Studies in Mathematics)
by Javier Duoandikoetxea
Hardcover: 222 Pages (2000-12-12)
list price: US$39.00 -- used & new: US$31.20
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Asin: 0821821725
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Fourier analysis encompasses a variety of perspectives and techniques. This volume presents the real variable methods of Fourier analysis introduced by Calderón and Zygmund. The text was born from a graduate course taught at the Universidad Autónoma de Madrid and incorporates lecture notes from a course taught by José Luis Rubio de Francia at the same university. Motivated by the study of Fourier series and integrals, classical topics are introduced, such as the Hardy-Littlewood maximal function and the Hilbert transform. The remaining portions of the text are devoted to the study of singular integral operators and multipliers. Both classical aspects of the theory and more recent developments, such as weighted inequalities, $H^1$, $BMO$ spaces, and the $T1$ theorem, are discussed. Chapter 1 presents a review of Fourier series and integrals; Chapters 2 and 3 introduce two operators that are basic to the field: the Hardy-Littlewood maximal function and the Hilbert transform. Chapters 4 and 5 discuss singular integrals, including modern generalizations. Chapter 6 studies the relationship between $H^1$, $BMO$, and singular integrals; Chapter 7 presents the elementary theory of weighted norm inequalities. Chapter 8 discusses Littlewood-Paley theory, which had developments that resulted in a number of applications. The final chapter concludes with an important result, the $T1$ theorem, which has been of crucial importance in the field. This volume has been updated and translated from the Spanish edition that was published in 1995. Minor changes have been made to the core of the book; however, the sections, "Notes and Further Results" have been considerably expanded and incorporate new topics, results, and references. It is geared toward graduate students seeking a concise introduction to the main aspects of the classical theory of singular operators and multipliers. Prerequisites include basic knowledge in Lebesgue integrals and functional analysis. ... Read more

6. Exercises in Fourier Analysis
by T. W. Körner
Paperback: 396 Pages (1993-09-24)
list price: US$50.00 -- used & new: US$43.33
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Asin: 0521438497
Average Customer Review: 5.0 out of 5 stars
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Fourier analysis is an indispensable tool for physicists, engineers and mathematicians. A wide variety of the techniques and applications of fourier analysis are discussed in Dr. Körner's highly popular book, An Introduction to Fourier Analysis (1988).In this book, Dr. Körner has compiled a collection of exercises on Fourier analysis that will thoroughly test the reader's understanding of the subject. They are arranged chapter by chapter to correspond with An Introduction to Fourier Analysis, and for all who enjoyed that book, this companion volume will be an essential purchase. ... Read more

Customer Reviews (1)

5-0 out of 5 stars Routine exercises studded with some gems
As the author notes in the Preface, this is an exercise book rather than a problem book; most of the exercises are further developments of ideas in the main text (Korner's book "Fourier Analysis"), and are not supposed to be hard. (The exercises that are hard, are liberally hinted and explained so you can still work through them without undue mental strain.)There are also many alternate proofs of results in the main text.

Nevertheless the book does contain many gems. These are primarily results that depend on some of the same ideas as Fourier Analysis without really being Fourier Analysis. Some of these gems: an "insufficiently asked question of Halmos" (whether there is a uniformly convergent sum that does not satisfy the Weierstrass M-test); a harmonic analysis experiment involving two potatoes and some string; a complete proof of Apery's theorem that the Riemann Zeta Function zeta(3) is irrational; the Feynman trick for evaluating integrals by differentiating under the integral sign; a proof of the Prime Number Theorem; and Karamata's proof of Littlewood's strengthening of Tauber's Theorem.

The index is full of jokes; be sure to read the index and browse any interesting-looking items. ... Read more

7. A First Course in Fourier Analysis
by David W. Kammler
Paperback: 864 Pages (2008-01-28)
list price: US$75.00 -- used & new: US$44.89
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Asin: 0521709792
Average Customer Review: 4.5 out of 5 stars
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This unique book provides a meaningful resource for applied mathematics through Fourier analysis. It develops a unified theory of discrete and continuous (univariate) Fourier analysis, the fast Fourier transform, and a powerful elementary theory of generalized functions and shows how these mathematical ideas can be used to study sampling theory, PDEs, probability, diffraction, musical tones, and wavelets. The book contains an unusually complete presentation of the Fourier transform calculus. It uses concepts from calculus to present an elementary theory of generalized functions. FT calculus and generalized functions are then used to study the wave equation, diffusion equation, and diffraction equation. Real-world applications of Fourier analysis are described in the chapter on musical tones. A valuable reference on Fourier analysis for a variety ofstudents and scientific professionals, including mathematicians, physicists, chemists, geologists, electrical engineers, mechanical engineers, and others. ... Read more

Customer Reviews (3)

5-0 out of 5 stars This was my favorite text in college.
I'm an electrical engineer, with a focus in signal processing.This is the book I learned Fourier analysis from, and once I did, the classes that EEs usually dread were relatively easy for me.This is the only textbook I actually read every chapter of (and we only covered the first half in the Fourier analysis course).Kammeler writes in a conversational style, which I like in a text, and goes through many practical examples in math, physics, and engineering.I appreciated the rigor devoted to generalized functions (Dirac deltas are almost always glossed over in engineering texts, and thus remain mysterious and sometimes non-sensical), yet Kammeler always keeps intuition close by so it's relatively easy to follow if you're not a mathematician.The parts I didn't like were when Kammeler fell back on more elementary yet more complicated presentations to avoid introducing too many new concepts.For example, I think the FFT is most easily understood with Z-transforms and multirate systems, and that Fourier analysis in general is more easily understood in terms of Hilbert spaces.It's hard to fault him for it though, because it's primarily a math book and needs to be mostly self-contained.It's also typeset in LaTeX, and looks beautiful.

3-0 out of 5 stars Not an accurate title
I used this book as part of a class at the University of Maryland.What I have discovered is that Kammler didn't really write a very good book for a first course in Fourier analysis.I am a math/physics major and found the book to be very scattered for a FIRST course.For example, the first chapter just dumps a whole bunch of information without presenting much background or context.That being said, I do think the book contains a lot of valuable information and might be good for students already familiar with Fourier analysis (I should note that I was familiar with Fourier series and Fourier transforms prior to the class).

5-0 out of 5 stars Moder Approach, Good Balance between Theory & Applications
I have been interested in the Mathematics of Fourier Series/Fourier Transform methods for well over 15 years. I own already well over 10 books on this subject. The book by David Kammler strikes me as having a particularly good balance between theory and applications as well as takinga modern computer approach to this ever relevant subject.Important topicssuch as sampling theory and the Fast Fourier Transform (FFT) are wellcovered and explained in detail. Also,chapters that apply FourierAnalysis to important physical areas (heat conduction, light diffraction,wave propagation, musical sound, etc.) illustrate and higlight therelevance of Fourier Methods in the real worls. There is also a nicesummary at the end of the book that explains the histoy and most importantapplication of Fourier Analyis (very nice). Ample computer excerices andthe traditional proof/derivation homework problems are included. The bookalso seems to prepare the reader well for the increasingly subject ofWavelets and applying them musical sound. Also, what makes the book standout from more traditional ones is the emphasis on Numerical Method andusing the computer to solve or illustrate some of the powers of FourierAnalysis. Readers considering using this text should best have a backgroundin calcus, differential equations and Matrix methods. This probably puts itat the junior/senior undergradudate level. 1st year graduate students mightalso benefit from the text.

In a nutshell this is an excellent textbookfor anyone serious about Fourier Analysis and applying those methods viacomputer (or pencil)to real world situation. This is probably one of thebest books yet on this very important subject. Highly Recommended! ... Read more

8. Modern Fourier Analysis (Graduate Texts in Mathematics)
by Loukas Grafakos
Hardcover: 507 Pages (2008-11-26)
list price: US$69.95 -- used & new: US$48.99
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Asin: 0387094334
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The primary goal of this text is to present the theoretical foundation of the field of Fourier analysis. This book is mainly addressed to graduate students in mathematics and is designed to serve for a three-course sequence on the subject. The only prerequisite for understanding the text is satisfactory completion of a course in measure theory, Lebesgue integration, and complex variables. This book is intended to present the selected topics in some depth and stimulate further study. Although the emphasis falls on real variable methods in Euclidean spaces, a chapter is devoted to the fundamentals of analysis on the torus. This material is included for historical reasons, as the genesis of Fourier analysis can be found in trigonometric expansions of periodic functions in several variables.

While the 1st edition was published as a single volume, the new edition will contain 120 pp of new material, with an additional chapter on time-frequency analysis and other modern topics. As a result, the book is now being published in 2 separate volumes, the first volume containing the classical topics (Lp Spaces, Littlewood-Paley Theory, Smoothness, etc...), the second volume containing the modern topics (weighted inequalities, wavelets, atomic decomposition, etc...).

From a review of the first edition:

“Grafakos’s book is very user-friendly with numerous examples illustrating the definitions and ideas. It is more suitable for readers who want to get a feel for current research. The treatment is thoroughly modern with free use of operators and functional analysis. Moreover, unlike many authors, Grafakos has clearly spent a great deal of time preparing the exercises.” - Ken Ross, MAA Online

... Read more

9. Fourier Analysis and Imaging
by Ronald Bracewell
Hardcover: 704 Pages (2004-01-31)
list price: US$179.00 -- used & new: US$97.79
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Asin: 0306481871
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As Lord Kelvin said, "Fourier's theorem is not only one of themost beautiful results of modern analysis, but it may be said tofurnish an indispensable instrument in the treatment of nearly everyrecondite question in modern physics." It has remained durableknowledge for a century and has extended its applicability to topicsas diverse as medical imaging (CT scanning), the presentation ofimages on screens and their digital transmission, remote sensing,geophysical exploration, and many branches of engineering. Fourier Analysis and Imaging is based on years of teaching acourse at senior or early graduate level and will also be a welcomeaddition to the reference library of those many professionals whosedaily activities involve Fourier analysis in its many guises. ... Read more

10. Classical Fourier Analysis (Graduate Texts in Mathematics)
by Loukas Grafakos
Hardcover: 492 Pages (2008-10-06)
list price: US$69.95 -- used & new: US$52.79
(price subject to change: see help)
Asin: 0387094318
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Editorial Review

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The primary goal of this text is to present the theoretical foundation of the field of Fourier analysis. This book is mainly addressed to graduate students in mathematics and is designed to serve for a three-course sequence on the subject. The only prerequisite for understanding the text is satisfactory completion of a course in measure theory, Lebesgue integration, and complex variables. This book is intended to present the selected topics in some depth and stimulate further study. Although the emphasis falls on real variable methods in Euclidean spaces, a chapter is devoted to the fundamentals of analysis on the torus. This material is included for historical reasons, as the genesis of Fourier analysis can be found in trigonometric expansions of periodic functions in several variables.

While the 1st edition was published as a single volume, the new edition will contain 120 pp of new material, with an additional chapter on time-frequency analysis and other modern topics. As a result, the book is now being published in 2 separate volumes, the first volume containing the classical topics (Lp Spaces, Littlewood-Paley Theory, Smoothness, etc...), the second volume containing the modern topics (weighted inequalities, wavelets, atomic decomposition, etc...).

From a review of the first edition:

“Grafakos’s book is very user-friendly with numerous examples illustrating the definitions and ideas. It is more suitable for readers who want to get a feel for current research. The treatment is thoroughly modern with free use of operators and functional analysis. Morever, unlike many authors, Grafakos has clearly spent a great deal of time preparing the exercises.” - Ken Ross, MAA Online

... Read more

11. Introduction to Fourier Analysis and Wavelets (Graduate Studies in Mathematics)
by Mark A. Pinsky
Hardcover: 376 Pages (2009-02-18)
list price: US$69.00 -- used & new: US$40.00
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Asin: 082184797X
Average Customer Review: 5.0 out of 5 stars
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This book provides a concrete introduction to a number of topics in harmonic analysis, accessible at the early graduate level or, in some cases, at an upper undergraduate level. Necessary prerequisites to using the text are rudiments of the Lebesgue measure and integration on the real line. It begins with a thorough treatment of Fourier series on the circle and their applications to approximation theory, probability, and plane geometry (the isoperimetric theorem). Frequently, more than one proof is offered for a given theorem to illustrate the multiplicity of approaches. The second chapter treats the Fourier transform on Euclidean spaces, especially the author's results in the three-dimensional piecewise smooth case, which is distinct from the classical Gibbs-Wilbraham phenomenon of one-dimensional Fourier analysis. The Poisson summation formula treated in Chapter 3 provides an elegant connection between Fourier series on the circle and Fourier transforms on the real line, culminating in Landau's asymptotic formulas for lattice points on a large sphere. Much of modern harmonic analysis is concerned with the behavior of various linear operators on the Lebesgue spaces $L^p(\mathbb{R}^n)$. Chapter 4 gives a gentle introduction to these results, using the Riesz-Thorin theorem and the Marcinkiewicz interpolation formula. One of the long-time users of Fourier analysis is probability theory. In Chapter 5 the central limit theorem, iterated log theorem, and Berry-Esseen theorems are developed using the suitable Fourier-analytic tools. The final chapter furnishes a gentle introduction to wavelet theory, depending only on the $L_2$ theory of the Fourier transform (the Plancherel theorem). The basic notions of scale and location parameters demonstrate the flexibility of the wavelet approach to harmonic analysis. The text contains numerous examples and more than 200 exercises, each located in close proximity to the related theoretical material. Originally published by Brooks Cole/Cengage Learning as ISBN: 978-0-534-37660-4. ... Read more

Customer Reviews (3)

5-0 out of 5 stars Excellent textbook and reference, which is readable!
I needed to learn Fourier Analysis and Wavelets, and this book is excellent as a textbook and as a reference.It is also quite readable.We need more mathematics books like this one.

5-0 out of 5 stars Fourier Analysis and Wavelets for Everyone
As a probabilist and statistician, with a Ph.D., having worked at universities, as a consultant, and in industry for approximately forty years, I had previously employed Fourier Analysis only as a tool, not having studied the subject as a discipline unto itself. Dr. Pinsky's book has allowed me to learn the subject more deeply, and from a different, exciting viewpoint. In addition, I needed a resource that would permit me to learn about wavelets. How wonderful to find a book that includes both topics! Moreover, this book is a pleasure to read, with pencil and paper, to work through the ideas. It is extraordinarily well-written, which is not surprising, given the clarity and excellence of Dr. Mark A. Pinsky's other works. Pinsky's grammar is excellent, which is extremely refreshing. Many modern authors cause me to believe that I should have a red pen to correct grammatical errors while reading their works. This book is much more than just a textbook; it is beautiful mathematically and beautifully written.

5-0 out of 5 stars For the Students!
Courses in harmonic analysis have a central place in the course offerings of every math department, be it pure or applied;-- and the subject is as important as ever! Yet it has not always been easy for an instructor to find a book that is right for the students. Some books might be too skimpy on proofs, or not deep enough.-- Or the applications may somehow be artificial, or contrived. Afterall, we teach the material to engineers!-- It is a relief to find, in Pinsky's lovely new book, a balanced approach to the subject. The motivation and the history receive a beautiful presentation, as do the technical points and proofs. And the historical comments- sprinkled throughout the book- bring the subject to life. At the same time, the book is forward looking, and it has been tested in courses. Great exercises! The structure of the exposition is friendly, and gently leads the reader toward the exciting new wavelet material in the last hundred or so pages of the book. The student thereby gets a sense of how the central questions in wavelet theory have their root in the more classical ideas of harmonic analysis. ... Read more

12. Fourier Analysis: An Introduction (Princeton Lectures in Analysis, Volume 1)
by Elias M. Stein, Rami Shakarchi
Hardcover: 320 Pages (2003-03-17)
list price: US$62.50 -- used & new: US$44.00
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Asin: 069111384X
Average Customer Review: 4.5 out of 5 stars
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This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when studying problems in the physical sciences--that an arbitrary function can be written as an infinite sum of the most basic trigonometric functions.

The first part implements this idea in terms of notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform; a clear introduction to the subject serves to avoid technical difficulties. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression.

In organizing their exposition, the authors have carefully balanced an emphasis on key conceptual insights against the need to provide the technical underpinnings of rigorous analysis. Students of mathematics, physics, engineering and other sciences will find the theory and applications covered in this volume to be of real interest.

The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Fourier Analysis is the first, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory. ... Read more

Customer Reviews (8)

5-0 out of 5 stars Nice book so far
I still have not read anything after chapter two, but the book look nice so far. It has a somewhat different approach by trying to avoid measure theory and still making a few comments on it for those who have already studied.

5-0 out of 5 stars excellent
I took many semesters of analysis in college as a math major, and I think I learned more useful knowledge from this book than from all those classes. Of course the classes helped prepare me to absorb what's in the book, but still it seems to me that the book strikes a good balance between generality and comprehensibility. Many of the books I used in school were too focussed on proving the most general version of every theorem, and failed to provide motivation or useful experience with the objects which the theorems actually describe. By taking fourier series as the motivating idea, the authors capture the historical spirit of the subject as well as that aspect of it which students are most likely to use in real work.

5-0 out of 5 stars very good but it is not an introduction
This is a very nice book in Fourier analysis with strong applications or examples in elementary partial differential equations. It is the first book of the three volumes set in the Princeton Lectures in Analysis. However, it is not an introductory text and some background in elementary analysis is required to fully appreciate its content.

5-0 out of 5 stars Excellent for an easy intro to distributions and it's applications
This is a somewhat biased review because sometimes I find myself searching for a good reference that treats a subject matter that is well-known in an easy, direct and accessible way. When I find such a book I end up relieved. This is what happened with the book by Stein and Shakarchi titled "Fourier Analysis".

In my case the search was for easy and accessible treatement of the theory of distributions in general and its applications to the wave equation in particular.

There are a number of references that treat this subject matter but all the ones I know of do this from a more advanced point of view. Stein and Shakarchi's book stems from an undergraduate lecture sequence thought at Princeton and the level of the text is indeed appropriate for the bright undergraduate who may or may not major in mathematics later on.

This is unlike PDE books by Taylor, or lecture notes by Melrose, or even the tiny booklet by Friedlander and Joshi that introduce distributions and their application to PDEs (like the wave equation) and certainly unlike Hörmanders comprehensive 4-volume treatment of the whole subject matter. All these references shoot significantly higher in terms of technical sophistication and I'd certainly not recommend them to typical engineering students for self-study. As possible exception I might mention Shubin's PDE books and encyclopedia contributions but they are more terse than the book under review and give less ground to more introductory matters.

Not so the book under review. It's an excellent, well-illustrated and clear presentation of the theory of distributions and its application to the wave equation, covering important (and old) techniques like the method of descend, which is still lacking from many contemporary engineering mathematics textbooks. Yet the book is written in a form and style to be accessible to a typical reader with engineering mathematics background while still being "modern" in it's mathematical language.

Hence I have recommended this book to many colleagues (and received enthusiastic reactions) as the only and at that quite excellent introduction ín know of to the theory of distribution, PDEs in that language and Fourier Analysis in that language that I trust to be accessible for non-specialists and as a gentle and non-threatening introduction to more technical texts.

3-0 out of 5 stars OK, but not a masterpiece
I taught an advanced undergraduate/beginning graduate class on Fourier transforms using this book as the text and wasn't thrilled. The selection of material seems uneven to me. For instance, there's a lengthy discussion of convergence issues for functions on the circle. Then apparently the authors became tired and basically restrict the treatment of the continuous case to Schwartz functions (which, of course, is insufficient for virtually every application). Also, the chapter on L_2 convergence seems to have been written on an off day.

These weaknesses don't make the book worthless, but in my opinion, there are better efforts on the market. One of my favorites for roughly this level of sophistication would be the book of Dym and McKean (which, admittedly, is a little more advanced). ... Read more

13. Introduction to Fourier Analysis on Euclidean Spaces. (PMS-32)
by Elias M. Stein, Guido Weiss
Hardcover: 312 Pages (1971-11-01)
list price: US$97.50 -- used & new: US$64.94
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Asin: 069108078X
Average Customer Review: 5.0 out of 5 stars
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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

Customer Reviews (1)

5-0 out of 5 stars 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 withelementary harmonic analysis, that is, this book is intended for graduatestudents and working mathematicians. Maybe some advanced undergraduatescould cover certain parts of the material.

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

The contents of the book are: The Fourier Transform; BoundaryValues of Harmonic Functions; The Theory of H^p Spaces on Tubes; SymmetryProperties of the Fourier Transform; Interpolation of Operators; SingularIntegrals and Systems of Conjugate Harmonic Functions; Multiple FourierSeries.

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

These three volumes should be present in every analyst'slibrary.

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

14. Fourier Analysis, Self-Adjointness (Methods of Modern Mathematical Physics, Vol. 2)
by Michael Reed, Barry Simon
Hardcover: 361 Pages (1975-10-12)
list price: US$166.00 -- used & new: US$69.99
(price subject to change: see help)
Asin: 0125850026
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This volume will serve several purposes: to provide an introduction for graduate students not previously acquainted with the material, to serve as a reference for mathematical physicists already working in the field, and to provide an introduction to various advanced topics which are difficult to understand in the literature. Not all the techniques and application are treated in the same depth. In general, we give a very thorough discussion of the mathematical techniquesand applications in quatum mechanics, but provide only an introduction to the problems arising in quantum field theory, classical mechanics, and partial differential equations. Finally, some of the material developed in this volume will not find applications until Volume III. For all these reasons, this volume contains a great variety of subject matter. To help the reader select which material is important for him, we have provided a "Reader's Guide" at the end of each chapter. ... Read more

15. Classical and Modern Fourier Analysis
by Loukas Grafakos
Hardcover: 870 Pages (2003-06-14)
list price: US$73.33 -- used & new: US$40.58
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Asin: 013035399X
Average Customer Review: 4.0 out of 5 stars
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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 Lp functions—one of the most difficult and celebrated theorems in Fourier Analysis. A complete Appendix contains a variety of miscellaneous formulae.Lp 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

Customer Reviews (1)

4-0 out of 5 stars New version (s) available
This is just to let the prospective buyer know that the book has been re-released in two volumes: Classical Fourier Analysis and Modern Fourier Analysis. ... Read more

16. Who Is Fourier?: A Mathematical Adventure
by Transnational College of LEX
Paperback: 435 Pages (1995-04-01)
list price: US$29.95 -- used & new: US$24.55
(price subject to change: see help)
Asin: 0964350408
Average Customer Review: 4.5 out of 5 stars
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In Who is Fourier? A Mathematical Adventure, the student authors take the reader along on their adventure of discovery of Fourier's wave analysis, creating a work that gradually moves from basics to the more complicated mathematics of trigonometry, exponentiation, differentiation, and integration. This is done in a way that is not only easy to understand, but is actually fun!

Professors and engineers, with high school and college students following closely, comprise the largest percentage of our readers. It is a must-have for anyone interested in music, mathematics, physics, engineering, or complex science.

Dr. Yoichiro Nambu, 2008 Nobel Prize Winner in Physics, served as a senior adviser to the English version of Who is Fourier? A Mathematical Adventure. ... Read more

Customer Reviews (42)

5-0 out of 5 stars Excellent intro to wave mechanics
I've taken many physics and engineering courses that use wave equations in one form or another.What they all lacked, however, was the overarching principles that tied wave equations in one discipline to wave equations in another.This book explains those connections and how one complicated wave equation can be broken down into its constituent parts.I highly recommend this book to those who love science but struggle with the math behind periodic systems.

4-0 out of 5 stars Who is fourier?
Excellente book. I don't have doubt to recommend this example for a dummies like me in complex but useful maths. I also suggest for a students of geophysics and engineer in general. They transform the difficult into a easy way to solve the problems.

4-0 out of 5 stars Very well written summary of Fourier analysis
I have read through most of the book.In general, this is an excellent introduction to the concept of Fourier analysis.It covers the basic concepts quite well.What I found quite interesting is that the authors of this book were able to take a relatively complicated idea and break it down and explain it in plain language, so that the reader developed an intuitive understanding of the concepts.The math in this book is a little heavy, and a familiarity with advanced algebra is necessary, and some background in calculus is helpful but not essntial because the authors do provide some background material in that area.

I used this book as a supplement to an engineering math course, and the explanations here were quite helpful to me in understanding the fundamental ideas.

My only complaint is that some of the cutesy stuff was a little distracting for me.When the authors stuck to straight language in explaining the material the book is at its best.Overall I highly recommend this book.

5-0 out of 5 stars Fourier made easy
One of the best math books for non PHD audience. I have a math degree myself and I have read a number of books covering Fourier transform. In all honesty this was the only book I really enjoyed. It had an easy to understand language, practical implications of the transform as well as background and meaning of Fourier coefficients. This book allowed me to develop the easiest and cleanest implementation of the Fourier transform.

5-0 out of 5 stars Who is Fourier ?
It is a Gem for anyone who wants to meet the spirit of fourier transforms. The approach is excellent and takes you off to the basics of fourier transforms smoothly.Books like these would surely get rid of the fears among students. I would suggest such books must be used as text books in high schools or even first year undergraduates to draw the interests of students ... Read more

17. Fourier Analysis on Finite Groups and Applications
by Audrey Terras
Paperback: 456 Pages (1999-05-01)
list price: US$58.00 -- used & new: US$52.09
(price subject to change: see help)
Asin: 0521457181
Average Customer Review: 4.5 out of 5 stars
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This book gives a friendly introduction to Fourier analysis on finite groups, both commutative and noncommutative. Aimed at students in mathematics, engineering and the physical sciences, it examines the theory of finite groups in a manner both accessible to the beginner and suitable for graduate research. With applications in chemistry, error-correcting codes, data analysis, graph theory, number theory and probability, the book presents a concrete approach to abstract group theory through applied examples, pictures and computer experiments. The author divides the book into two parts. In the first part, she parallels the development of Fourier analysis on the real line and the circle, and then moves on to analog of higher dimensional Euclidean space. The second part emphasizes matrix groups such as the Heisenberg group of upper triangular 2x2 matrices with 1's down the diagonal and entries in a finite field. The book concludes with an introduction to zeta functions on finite graphs via the trace formula. ... Read more

Customer Reviews (3)

4-0 out of 5 stars Innovative choice of topics, if somewhat sketchy in the presentation
The book presents a wealth of facts regarding finite structures related with the Fourier transform and its many applications - not only the usual stuff on the fast or discrete Fourier transforms (FFT/DFT), but also on Cayley graphs, error-correcting codes, group representation, and the Selberg trace formula, among other things (e.g., an introduction to the idea of random walks on groups).However, while I liked the choice of subjects and the book 'hits the point' many times, sometimes you get lost with the somewhat sketchy presentation, as if it has been assembled from slide presentations and unstructured lecture notes.The book also has some typos like non-matching parentheses, spacing errors, and dislocated super- and subscripts, most of them harmless.

I recommend the following textbooks as companion reading: the excellent and highly entertaining "Number Theory in Science and Communication," by Manfred R. Schroeder, and the basic "Applied Abstract Algebra" text by Rudolf Lidl and Günter Pilz. In Schroeder's book you will find a more structured presentation of a large subset of the subjects presented by Terras, while the abstract algebra that you may eventually need you will almost certainly find in the pedagogical exposition of Lidl & Piltz.

My rate: 5 stars for the innovative choice of topics and -1 star for the sketchy presentation, sometimes a little weird notation, and typos.This book deserves a better editorial and typographical treatment (as well as a hardcover) in a future (ideally updated) edition.

5-0 out of 5 stars To the point!
For students and users who need the facts! Perhaps they have
come accross a group problem in programming, in physics, in some course or other, or in a research assignment. Over the years, when teaching, I am often asked by students for directions to the facts of groups(usually finite) and their harmonic analysis.
And too often, I have had to send the poor student
to a multi-volume book set on the general theory. Sure much of it can be specialized to what the case demands. But the user[in e.g., programming, algorithms, fast transforms, error-correction codes, crystals, symmetry, quantum theory, engineering...]
typically isn't ready for the big picture yet, and more often than not, she will need to first look for a gentle introduction--just the facts!-- and written in a delightful style! The one that this author has perfected in her other books. Thanks! Now I do feel good about sending my students to this lovely little book. Great book! -- And gentle on my student's budget too.

5-0 out of 5 stars Great cross-fertilization book
This is a delightful book that covers broad areas of theoretical and practical mathematics from the standpoint of Group Theory and Fourier Analysis. Although Fourier Analysis is first in the title, I think Finite Groups should come first. The author takes the discrete perspective of traditionally continous functions of classical physics and other applications - which not suprisingly brings finite groups into the picture.This includes the applications of quadratic residues and primitive roots to areas not usally touched by number theory - such as the ceilings of concert halls. This is a fun book, and it doesn't pretend to be comprehensive or complete with regards to its choosen subjects. I'd call the book "A Grand Tour of Finite Groups and Fourier Analysis with Applications" if I were the publisher.The price is suprisingly affordable given the subject where new books usually cost O($100). ... Read more

18. Fourier Analysis on Number Fields (Graduate Texts in Mathematics) (v. 186)
by Dinakar Ramakrishnan, Robert J. Valenza
Hardcover: 350 Pages (1998-12-07)
list price: US$99.00 -- used & new: US$66.00
(price subject to change: see help)
Asin: 0387984364
Average Customer Review: 4.0 out of 5 stars
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The general aim of this book is to provide a modern approach to number theory through a blending of complementary algebraic and analytic perspectives, emphasizing harmonic analysis on topological groups. The more particular goal is to cover John Tate's visionary thesis, giving virtually all of the necessary analytic details and topological preliminaries--technical prerequisites that are often foreign to the typical, more algebraically inclined number theorist. While most of the existing treatments of Tate's thesis are somewhat terse and less than complete, the authors' intent is to be more leisurely, more comprehensive, and more comprehensible. The text addresses students who have taken a year of graduate-level courses in algebra, analysis, and topology. While the choice of objects and methods is naturally guided by specific mathematical goals, the approach is by no means narrow. In fact, the subject matter at hand is germane not only to budding number theorists, but also to students of harmonic analysis or the representation theory of Lie groups. Moreover, the work should be a good reference for working mathematicians interested in any of these fields. Specific topics include: topological groups, representation theory, duality for locally compact abelian groups, the structure of arithmetic fields, adeles and ideles, an introduction to class field theory, and Tate's thesis and applications. ... Read more

Customer Reviews (3)

5-0 out of 5 stars An Exellent Introductory Textbook to Modern Number Theory and Automorphic Forms
I agree very much on what Stephen Miller said. This textbook is a very execellent introductory textbook to modern number theory. It does not require any particular math background besides elementary undergraduate maths so that it is suitable to new graduate students. The exercises are very nice and helpful. The level is a little bit challenging. I ever taught courses based on this book twice and both students and I benefit a lot.

For the contents, the textbook provides a thourough treatment on basics of modern NT such as local fields, adeles, ideles, Fourier inverse formula etc. Moreover, I think the textbook might be the best source so far I know for on Tate's thesis as a textbook. It is a perfect starting book for readers who are interested on automorphic forms. Also, just as Miller said, it is also a good reference book to mathematicians with various background, not just merely number theorists.

So I recommend this textbook strongly.

Song Wang, the Morningside Center of Mathematics, AMSS, CAS, China.

2-0 out of 5 stars one more opinion
I don't agree with the previous reviewer about the value of this
book - I think that with several minor exceptions there is nothing in this book which could justify its publication.
Of course, as it is clear to every expert, there is nothing
really new in this book; but sometimes one can rewriteold
things in such a way that a new book is justified.
With the material of this book I know much better expositions
of every chapter of it (including harmonic analysis, number theory and Tate-Iwasawa method) in other sourses.
There are also some mistakes and errors (for example,
the Poisson summation formula is not proven),
some of which may cause the reader
think that there were mistakes on the original works.

This text could have appeared online as lecture notes,
but the publication of it by Springer confirm the well known fact of degradation of their mathematical series.

D. Ziegler

5-0 out of 5 stars A treat for beginners to some exciting areas of mathematics
"Fourier Analysis on Number Fields" provides a much-needed graduate text for number theorists and group theorists.Though necessarily difficult in partsbecause of the complicated material it covers, it isvery manageable for a student.It includes a number of exercises at theend of each of its seven chapters.At the same time, it is very valuablefor a researcher.Perhaps its best feature are the wonderful introductionsto each chapter.These give insightful historical overviews, in keepingwith the authors' theme of presenting material from disparate sourcestogether in a coherent text. It is obvious that they spent a lot ofattention on the beginner's needs.

Indeed, existing texts cover most ifnot all of the material in this new book.Others, including some new bookson automorphic forms, take the reader much further.However, not everyonehas the same starting point and all of these can be very frustrating for abeginner.The novelty and utility in this book is that it does not assumethe reader comes from some particular background.Off-hand I could namefive or six other books I would consult to learn the material"FANF" covers.But each comes from a different community ofmathematicians, with their own jargon, in different eras, and are intendedfor different audiences."FANF" sacrifices some proofs forclarity, and gives references to the classical sources for furtherdetails.

One of the authors' goals was to give explicit background on thestructure of the fields involved, particularly the delicate arithmeticstructure of number fields which is sometimes frustrating to learn fromother sources.They have covered the structure of locally-compact fieldsvery well and clearly.In fact, in one of our graduate courses at YaleUniversity last fall, lectures on p-adic groups and trees were based out ofthe presentation in "FANF."The book is very concrete, which isespecially useful for analysts who aren't used to doing integrals over,say, function fields in finite characteristic.I think it will be afavorite amongst this community - it treats advanced stages of "mathphobia."

At the same time this is the natural book for anintroductory course on modern automorphic forms.It completely covers theGL(1) theory and leaves the reader in an excellent position to continue onto study the Jacquet-Langlands theory.It has a nice treatment ofL-functions, and even includes some analytic results which featureprominently in the recent research of one of the authors.There isn't abook that I know of which fits the nice "FANF" occupies, andbetter yet, it complements the earlier ones very well.

Let me justmention two examples of recent research which explain why I think a bookcovering its various topics is so important.Hyman Bass and Alex Lubotzkyfound a counter-example to the Platonov conjecture.This problem involvesthe representation theory of profinite groups, and lattices acting ontrees."FANF" has beautiful treatments of these.At the sametime, a key ingredient of their proof was understanding the cohomology ofdiscrete subgroups of Lie groups.Ultimately this can be interpreted as aproblem in automorphic forms!In fact, they used results of David Voganand Gregg Zuckerman about cohomological representations in their work. Another example is that the "Selberg Property-Tau" has becomevery important in p-adic group theory; it originated as a bound on Laplaceeigenvalues in modular forms.Fortunately these aspects of algebraicgroups are becoming more deeply linked, and "FANF" is amost-recommended book to start learning any of these subjects from.

Stephen D. Miller Department of Mathematics Yale University ... Read more

19. A First Course in Wavelets with Fourier Analysis
by Albert Boggess, Francis J. Narcowich
Hardcover: 336 Pages (2009-09-08)
list price: US$99.95 -- used & new: US$63.08
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Asin: 0470431172
Average Customer Review: 5.0 out of 5 stars
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A comprehensive, self-contained treatment of Fourier analysis and wavelets—now in a new edition

Through expansive coverage and easy-to-follow explanations, A First Course in Wavelets with Fourier Analysis, Second Edition provides a self-contained mathematical treatment of Fourier analysis and wavelets, while uniquely presenting signal analysis applications and problems. Essential and fundamental ideas are presented in an effort to make the book accessible to a broad audience, and, in addition, their applications to signal processing are kept at an elementary level.

The book begins with an introduction to vector spaces, inner product spaces, and other preliminary topics in analysis. Subsequent chapters feature:

  • The development of a Fourier series, Fourier transform, and discrete Fourier analysis

  • Improved sections devoted to continuous wavelets and two-dimensional wavelets

  • The analysis of Haar, Shannon, and linear spline wavelets

  • The general theory of multi-resolution analysis

  • Updated MATLAB® code and expanded applications to signal processing

  • The construction, smoothness, and computation of Daubechies' wavelets

  • Advanced topics such as wavelets in higher dimensions, decomposition and reconstruction, and wavelet transform

Applications to signal processing are provided throughout the book, most involving the filtering and compression of signals from audio or video. Some of these applications are presented first in the context of Fourier analysis and are later explored in the chapters on wavelets. New exercises introduce additional applications, and complete proofs accompany the discussion of each presented theory. Extensive appendices outline more advanced proofs and partial solutions to exercises as well as updated MATLAB® routines that supplement the presented examples.

A First Course in Wavelets with Fourier Analysis, Second Edition is an excellent book for courses in mathematics and engineering at the upper-undergraduate and graduate levels. It is also a valuable resource for mathematicians, signal processing engineers, and scientists who wish to learn about wavelet theory and Fourier analysis on an elementary level. ... Read more

Customer Reviews (4)

5-0 out of 5 stars A necesary textboox
A wonderful book. Here the different topics are treated carefully in a very straightforward way. It gives a clear path for the real understanding of the wavelet analysis.

5-0 out of 5 stars Learn Wavelet
If you want to learn Wavelet theory in a easy way like reading a story book then this is the book. It deals with the most complicated thing in the easiest way.

5-0 out of 5 stars Informative Book for Wavelet Analysis!!!!
...this book is one of most informative and legible books on wavelt theories and applications.

The author paves the theoretical development about wavelets and multi-resolution analysis EXCELLENTLY. With this book, you can construct wavelets for your own applications in engineering and science disciplines.

This book is very good for first year engineering-majored graduate students and all engineering scholars.

4-0 out of 5 stars A good Starter on Wavelets
At the time of writing of this review (October 2001), a standard academic search procedure
produces about twenty references per week of scientific papers using wavelet analysis in a very wide spectrum of sciences. More than 160 english language books have been published on wavelets since the first books appeared around 1990. Yet even now it is rare to find a book on this subject which is aiming at undergraduate students and yet is mathematically responsible, without being heavy going. Boggess and Narcovich have tried to do just that, and to my mind have admirably succeeded.
Assuming a standard background knowledge in calculus and linear algebra that many science and engineering students acquire in their first two years at university, they present the basics of Fourier analysis and wavelets in eight brief chapters. To prepare the way, they start in chapter 0 with an introduction to inner product spaces, without using advanced analysis, and building on the experience with ordinary vector spaces.
Also a sniff of linear operator theory is offered.
Chapter 1 introduces Fourier series in real and complex form. These originated in the eighteenth century study of vibrations and in the theory of heat, made famous by Fourier's classic book of 1808: Analytical Theory of Heat. The mathematical claims Fourier made, but which he could not all prove himself, gave the impetus to an enormous development of both mathematical theory and applications in all fields of natural science, which is still going on today. The applications briefly mentioned here are denoising and compression of signals, and finding the solution of partial differential equations. Various aspects of the convergence of Fourier series are dealt with. All concepts are illustrated with a good set of clear figures, and the chapter finishes with exercises that are going from very elementary to a little more ambitious, sometimes involving the use of simple computer algebra tasks. This format is maintained thorugh the entire text, except for the last chapter.
Chapter 2 proceeds with the Fourier Transform, including the important theory of linear time invariant filters. The existence of the impulse response function and its convolution character are shown. As an example the noise reducing Butterworth filter is presented. Sampling and the Nyquist frequency are touched upon, and a derivation of the uncertainty relations, originally coming from quantum mechanics, is given.
To analyse discrete data, one needs the discrete Fourier Transform, which is the subject of chapter 3, including of course the Fast Fourier Transform. Also the z-transform is introduced. Examples given are elementary cases of parameter identification in vibration, numerical solution of ordinarydifferentialequations, as well as in the exercises: noise reduction and data compression.

These first 153 pages serve as a good undergraduate introduction to Fourier analysis.
The second half of the book is devoted to wavelets. Chapter 4 deals exclusively with Haar wavelets which are the oldest wavelets because they date from 1910! These wavelets constitute an orthonormal basis of functions, which makes for fast calucation, a very important aspect for many applications. The core ideas of the central concept of a "multiresolution analysis" of a signal, can be demonstrated with these simple wavelets. All of this is already understandable without the machinery of the preceding Fourier analysis, so you could jump into the book here and start reading about wavelets right away, picking up the Fourier analysis from the first part bit by bit as the need arises. As applications denoising and compression are mentioned again, as is the detection of a discontinuity in a signal.
The general case of a multiresolution analysis is the subject ofchapter 5. Again a large part of the discussion can be swallowed without the need of the Fourier transform point of view. The explanation of the structure of a multiresolution analysis leading to an orhtonormal basis of wavelets is straightforward and clear. It is only when we want to go into more detail about the precise characteristics of the underlying wavelet and scaling function that the Fourier point of view is introduced. This then leads up to the presentation of the famous Daubechies wavelets in chapter 6. These wavelets revolutionised the field after their publication in 1988.
Chapter 7 which closes the book, gives several short remarks about various other topics among which are two-dimensional wavelets, and the continuous wavelet transform.
This chapter is more sketchy than the others, and left me much less satisfied. Also the motivation why these subjects are chosen was lacking almost completely, and there are no exercises. I was particularly disappointed not to find any discussion of the relative merits of the continuous versus the discrete wavelet transform, and there is no mention of any application of the continuous case. Yet the latter is also used frequently in many important scientific applications, and it started the modern wavelet endeavour in the early eighties in France.
That being said I still think this is a very useful book for anybody wanting to start withwavelets at an undergraduate level. A few helpful Matlab Codes are collected in an appendix as well as the more difficult parts of some proofs.The exercises make this good course material, but as a text for self study it will also be quite satisfactory for many newcomers that find most of the existing books too demanding. ... Read more

20. Fourier Analysis
by James S. Walker
Hardcover: 462 Pages (1988-07-14)
list price: US$150.00 -- used & new: US$49.98
(price subject to change: see help)
Asin: 0195043006
Average Customer Review: 5.0 out of 5 stars
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Fourier analysis is a mathematical technique for decomposing a signal into identifiable components.It is used in the study of all types of waves.This book explains the basic mathematical theory and some of the principal applications of Fourier analysis in areas ranging from sound and vibration to optics and CAT scanning.The author provides in-depth coverage of the techniques and includes exercises that demonstrate straightforward applications of formulas as well as more complex problems. ... Read more

Customer Reviews (1)

5-0 out of 5 stars Fourier Analysis
Fourier Analysis book by James Walker is one of the finest books I have read thus
far on Fourier's work.It is well written and supplemented with examples with a
profound explanation.I have read and worked through many books on Forurier series and
transforms but have not found any book that does as well as that by Walker.This
book is unique and should serve as excellent source of information for people in math
and physics. ... Read more

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