Product Description
This book is intended as an introduction to classical Fourier analysis, Fourier series, the Fourier transform, and Fourier sine and cosine transforms. The topics are developed slowly for the reader who has never seen them before, with a preference for clarity of exposition over elegance in stating and proving results. More recent developments, such as the discrete and fast Fourier transforms and wavelets, are covered in the last two chapters, where the approach is elementary, intended for the novice reader. The first three, short, chapters present requisite background material, and these could be read as a short course in functional analysis. The text includes many historical notes to place the material in a cultural and mathematical context; from the fact that Jean Baptiste Joseph Fourier was the nineteenth, but not the last, child in his family to the impact that Fourier series have had on the evolution of the concept of the integral. ... Read more

Customer Reviews (2)

Very good except for some errors
The book is recomendable but it has got several errors. For example, the hint of the exercise 4.7 is not such "hint". Anymore found this or other errors?

Anyway, the exposition is clear and ordered and you will get a great insight on Fourier analyisis.

Great introduction, but clearly a first edition
I'm just finishing a semester-long course which uses this book as an outline and as a source for homework questions.Anyone with basic knowledge of analysis will have no problem picking up this book and learning a great deal about Fourier decomposition and transformations, but there are a few problems with it.The questions at the end of each section don't always make it clear exactly what is to be answered and what is to be assumed.Also, there have been a few questions that neither I, my classmates, nor my professor could solve, which leads me to believe that there are fundamental problems with a few of the questions (I could be wrong though ;) )Finally, the last chapter seems to use different notation for some things than is used in the rest of the book, so be alert.Overall, the clarity of the material outweighs these flaws and if you have any interest in REALLY understanding Fourier and Wavelet series, get this book. ... Read more

Product Description
"D. Walnut's lovely book aims at the upper undergraduate level, and so it includes relatively more preliminary material...than is typically the case in a graduate text. It goes from Haar systems to multiresolutions, and then the discrete wavelet transform... The applications to image compression are wonderful, and the best I have seen in books at this level. I also found the analysis of the best choice of basis and wavelet packet especially attractive. The later chapters include MATLAB codes. Highly recommended!"--- Bulletin of the AMS

"[This text] is carefully prepared, well-organized, and covers a large part of the central theory...[there are] chapters on biorthogonal wavelets and wavelet packets, topics which are rare in wavelet books.Both are important, and this feature is an extra argument in favour of [this] book...the material is accessible [even] to less advanced readers...the book is a nice addition to the series." --Zentralblatt Math

"This book can be recommended to everyone, especially to students looking for a detailed introduction to the subject." --Mathematical Reviews

"This textbook is an introduction to the mathematical theory of wavelet analysis at the level of advanced calculus.Some applications are described, but the main purpose of the book is to develop---using only tools from a first course in advanced calculus---a solid foundation in wavelet theory. It succeeds admirably.... Part I of the book contains 112 pages of preliminary material, consisting of four chapters on 'Functions and Convergence,' 'Fourier Series,' 'Fourier Transforms,' and 'Signals and Systems....' This preliminary material is so well written that it could serve as an excellent supplement to a first course in advanced calculus.... The heart of the book is Part III: 'Orthonormal Wavelet bases.'This material has become the canonical portion of wavelet theory.Walnut does a first-rate job explaining the ideas here.... Ample references are supplied to aid the reader.... There are exercises at the end of each section, 170 in all, and they seem to be consistent with the level of the text....To cover the whole book would require a year.An excellent one-semester course could be based on a selection of chapters from Parts II, III, and V." --SIAM Review

"An Introduction to Wavelet Analysis" provides a comprehensive presentation of the conceptual basis of wavelet analysis, including the construction and application of wavelet bases.

The book develops the basic theory of wavelet bases and transforms without assuming any knowledge of Lebesgue integration or the theory of abstract Hilbert spaces. The book elucidates the central ideas of wavelet theory by offering a detailed exposition of the Haar series, and then shows how a more abstract approach allows one to generalize and improve upon the Haar series. Once these ideas have been established and explored, variations and extensions of Haar construction are presented. The mathematical prerequisites for the book are a course in advanced calculus, familiarity with the language of formal mathematical proofs, and basic linear algebra concepts.

Features:

* Rigorous proofs have consistent assumptions about the mathematical background of the reader (does not assume familiarity with Hilbert spaces or Lebesgue measure).

* Complete background material is offered on Fourier analysis topics.

* Wavelets are presented first on the continuous domain and later restricted to the discrete domain for improved motivation and understanding of discrete wavelet transforms and applications.

* Special appendix, "Excursions in Wavelet Theory," provides a guide to current literature on the topic

* Over 170 exercises guide the reader through the text.

"An Introduction to Wavelet Analysis" is an ideal text/reference for a broad audience of advanced students and researchers in applied mathematics, electrical engineering, computational science, and physical sciences. It is also suitable as a self-study reference for professionals. ... Read more

Customer Reviews (1)

From Fourier to wavelets to applications.
I take it as a healthy sign when there is a burst of new books in a sub-area of math. In wavelet analysis and its applications, we have seen a number of recent books arrive to university bookstores. Surprisingly there doesn't in fact seem to be much of an overlap of subject or scope, from one book to the next. The subject is infinite in many directions, for example the kind of student it is aimed at, the level, the specialized area within math itself, and the kind of application it is stressing. D. Walnut's lovely book aims at the upper undergraduate level, and so it includes relatively more preliminary material, for example Fourier series, than is typically the case in a graduate text. It goes from Haar systems to multirelutions, and then the discrete wavelet transform, starting on page 215. The applications to image compression are wonderful, and the best I have seen in books at this level. I also found the analysis of the best choice of basis, and wavelet packet, especially attractive. The later chapters include MATLAB codes.-- Highly recommended! ... Read more

Product Description
Fourier Analysis and Boundary Value Problems provides a thorough examination of both the theory and applications of partial differential equations and the Fourier and Laplace methods for their solutions. Boundary value problems, including the heat and wave equations, are integrated throughout the book. Written from a historical perspective with extensive biographical coverage of pioneers in the field, the book emphasizes the important role played by partial differential equations in engineering and physics. In addition, the author demonstrates how efforts to deal with these problems have lead to wonderfully significant developments in mathematics.
A clear and complete text with more than 500 exercises, Fourier Analysis and Boundary Value Problems is a good introduction and a valuable resource for those in the field.

Key Features
* Topics are covered from a historical perspective with biographical information on key contributors to the field
* The text contains more than 500 exercises
* Includes practical applications of the equations to problems in both engineering and physics ... Read more

Customer Reviews (5)

Having a professor who writes his own text not always good.
I had Professor Gonzalez for my applied math class and we used this book.while is a unique take, including history and background, the subject matter remained unclear at times. It is hard to grasp as a sophmore in college and it goes to fast in a classroom setting. While I agree that other professor's should use this book as a reference tool, I have experienced it first-hand in the classroom and it caused more confusion than anything else.

The best book onPDEs
I was confused with so many books on PDEs and found this book God sent.If you ask me buy this book if you are looking fora serious book on PDEs with lucid explanations and problem solving techniques this is the book. Though the answers for the problems aren't given, you will love doing problems, I studied this book for my PhD quals.Enjoy it, savor it.

Fourier Analysis and Boundary Value Problems
This is an excellent introduction to Fourier series and its
applications toboundary value problems. It has a very good combination of history, theory,and exercises. In particular,
the book is an ideal text for math students who are taking
their first PDE course and do not have enough backgound in physics. The only problem with this book is its high price.

One of "THE" Books on Fourier and Boundary Value Problems
This book is very complete and comprehensive. It is meant for the advanced undergraduate (and even graduate) student. The book states it assumes you know the 1st year of calculus and how to solve some ordinary differential equations. It would help if you took one more upper division math class, as to be familiar with the proofs.

In a nutshell, this book is great! I used it to study for my PhD Quals in physics. It has a nice portion on Green's functions.The book is 'mathematical' but the backdrop on the topics are based on physical problems. The historical notes are great, and act as a complement to the physical problems (I'm not a big fan of history in textbooks, but this book does a nice job of integrating it).

Lots of problems are given, though no solutions are given for any of them. It would be a nice compliment to at least show the start, a tid bit in the middle, and then the solution. The solution at the very least.There are other books that do provide solutions to the types of problems listed here, but it would be nice just to carry around this one when learning boundary value problems.

A must for anyone that works on any problems that have ODEs, PDEs, Fourier and Laplace transforms, Boundary Value Problems and Green's Functions.You won't be disappointed!

A comprehensive book on Fourier Analysis, PDE and BVP
This is one of the best book from which to learn the subject, as it covers all the topics with great clarity and includes historical developement which makes for great reading.The coverage is mathematical rigorous but yet well-written enough that an advance undergrad. would have no problem in learning the material for the 1st time from this book.The only drawback to this book is that there is no answer for any exercise (no solution manual available either).Learn the mathematics here and use other books for exercise. ... Read more

Product Description
Before applying multigrid methods to a project, mathematicians, scientists, and engineers need to answer questions related to the quality of convergence,whethera developmentwill pay out, whether multigrid will work for a particular application,and what the numerical properties are. Practical Fourier Analysis for Multigrid Methods uses a detailed and systematic description of local Fourier k-grid (k=1,2,3) analysis for general systems of partial differential equations to provide a framework that answers these questions.

This volume contains software that confirms written statements about convergence and efficiency of algorithms and is easily adapted to new applications. Providing theoretical background and the linkage between theory and practice, the text and software quickly combine learning by reading and learning by doing. The book enables understanding of basic principles of multigrid and local Fourier analysis, and also describes the theory important to those who need to delve deeper into the details of the subject.

The first chapter delivers an explanation of concepts, including Fourier components and multigrid principles. Chapter 2 highlights the basic elements of local Fourier analysis and the limits to this approach. Chapter 3 examines multigrid methods and components, supported by a user-friendly GUI. Chapter 4 provides case studies for two- and three-dimensional problems. Chapters 5 and 6 detail the mathematics embedded within the software system. Chapter 7 presents recent developments and further applications of local Fourier analysis for multigrid methods. ... Read more

Product Description
From the reviews: "Volume 1 covers a basic course in real analysis of one variable and Fourier series. It is well-illustrated, well-motivated and very well-provided with a multitude of unusually useful and accessible exercises. (...) There are three aspects of Courant and John in which it outshines (some) contemporaries: (i) the extensive historical references, (ii) the chapter on numerical methods, and (iii) the two chapters on physics and geometry. The exercises in Courant and John are put together purposefully, and either look numerically interesting, or are intuitively significant, or lead to applications. It is the best text known to the reviewer for anyone trying to make an analysis course less abstract. (...)" The Mathematical Gazette (75.1991.471 ... Read more

Customer Reviews (10)

Solutions to problems and exercises
Springer have reprinted the original 1960s Wiley editions of "Introduction to Calculus and Analysis" volumes I and II by Courant and John in three new volumes under their "Classics in Mathematics" title: "Introduction to Calculus and Analysis I (pages 1-661)" (ISBN: 3-540-65058-X), "Introduction to Calculus and Analysis II/1, Chapters 1-4 (pages 1-542)" (ISBN: 3-540-66569-2), and "Introduction to Calculus and Analysis II/2, Chapters 5-8 (pages 543-954)" (ISBN: 3-540-66570-6).The back section of Volume II/2 (pages 821-939) has solutions to the exercises in both the books comprising volume II, that is "Introduction to Calculus and Analysis II/1" and "Introduction to Calculus and Analysis II/2".

Note that when Volume I of the original Courant and John "Introduction to Calculus and Analysis" was published in the 1960s by Wiley, an accompanying solutions manual for Volume I was prepared by Prof. Albert A. Blank.When Volume II was published by Wiley, Prof. Blank's solutions were incorporated into the back of Volume II (in other words, Volume II comes with the answers to the questions at the back of the book... or in the back of Volume II/2 in the case of this Springer "Classics in Mathematics" reprint.)However, the Springer reprint of Wiley's Volume I lacks solutions to the exercises in the textbook.

If you buy Volume I, do a check on the Internet for an old 1960s copy of Prof. Albert Blank's "Problems in Calculus and Analysis", which is the original solutions manual to Courant's Volume I.

An Excellent Book
This (and Vol. 2) are both really good books that anyone with an interest in mathematics should own.Is it as good as Apostol's two volumes (Tommy I and II)?No.Is it as good as Spivak's "Calculus"?No.But it is still very good because the exposition is wonderful.I own both volumes and am glad - but if you only want one Courant book, please buy "What is Mathematics".

a superb book
This is a rewrite of the great book by Courant, and it does justice to its origin.I prefer the somewhat more charming original book of Courant myself, but I have taught from this one too and learned something more.

Since the original Courant costs $120 for the 2 volume set, this volume at $33 is a bargain, so snap it up.This is 10 times as valuable as most current $130 calculus books.

More than an introduction
Those books (volumes 1-2) can be seen as a new edition of Courant's classical Differential and Integral Calculus, volumes 1-2 (that can still be used for general calculus courses). The first volume was written while Courant was still alive, and the second was postumous. I believe that they are the best work to start understanding analysis. Indeed, for the general scientist (as a physicist) it contains all the theory needed for any application. The book is not easy reading though. Much of the text can be understood on first reading, but there are pretty profound sections, mostly on the appendixes, that turn the book genuinely onto a book of analysis. The second volume requires some mathematical maturity, and I doubt whether it is suitable for beginners, but it is simply the best book of multivariate calculus that I know - and it is really difficult to think of a better presentation. Courant was a giant, and his concept of mathematics shines in every page of those books (although he did not see the publication of the second volume, his hand can be seen in every page). For the serious mathematician, a must-have. For the beginner, the best way to get in love. Courant and John don't lie, they give every proof and guide you most gently in this complicated garden called mathematics. I'd give it aleph stars if it was possible.

Superior as an introductory calculus text!
I don't use the word "superior" lightly, but this book definitely warrants it.Courant was a first rate teacher and mathematician, and his brilliance shows in his exposition.The main obstacle to some readers may be that Courant does not follow the "cookbook calculus" approach that seems so rampant today, but actually bothers to prove his results.He does, however, reserve most of the more difficult proofs for the appendices at the end of the chapter, which is most appreciated.

The result is an exciting read, yet rigorous.The reader is very well prepared for future courses in mathematical analysis, and even has a leg up on real analysis.While Courant's insistence on proof does mean that the student needs to have a basic grounding in proof methods, this is usually a standard part of the undergraduate curriclum.Courant rightly recognizes that calculus should be taught in a logical, yet rigorous presentation from the beginning.The absence of this in modern texts mean that students learn how to manipulate formulas, but have no idea what makes the results they are assuming true.The "mechanics" of calculus and analysis, the most crucial thing to be learn, is missed.

In particular, I enjoyed his presentation of integration *before* differentiation, which goes against the grain of basic calc texts, yet is historically and pedagogically correct.Integration actually paves the way for differentiation, and gives more motivation for the FTC.Most texts on real analysis work in that order anyway, as an understanding of Lebesgue measure and integration is crucial to understanding the process of differentiation.

In addition, I don't think I have ever before or since seen such a careful explanation of the theory of the logarithm or exponential functions.Again, the presentation makes it work, as just introducing the "exponential function", then a little later, the "log function" as the "inverse" of the exponential function is, to put it mildly, artificial and distasteful.The natural progression from the definite integral definition of the logarithm to the exponential function is displayed in its full glory.

In short, Courant manages to present some of the most crucial results of calculus and basic analysis without boring the reader to tears with arcane details, or worse, leaving the reader hanging on important theorems and ideas.This is a balance only a great mathematician could strike, and it is clear why this book remains a classic after almost 60 years.

Note: The second volume of this work covers the multivariable portion of calculus, and will be more difficult to follow without prior exposure to the subject.However, the introductions to the theory of matrices and the calculus of variations are very readable, and it is recommended that the reader take the time to peruse them.Also, don't miss the material on special functions, lightly touched on in the first volume, but explained in fuller detail in the second. ... Read more

Product Description

From the reviews: "[…] the interested reader will find in Bremaud’s book an invaluable reference because of its coverage, scope and style, as well as of the unified treatment it offers of (signal processing oriented) Fourier and wavelet basics." Mathematical Reviews

Customer Reviews (1)

mathematically rigorous approach to signal processing
If you don't like the sloppy arguments often provided by engineering books, then this is the book for you!The prerequisite is some familiarity with the Lebesgue Integral.The author provides a clean development of the L1 and L2 theory for both Fourier Series and the Fourier Transform.In addition, he covers the basics of signal processing and wavelets in a concise, but rigorous, style. ... Read more

Customer Reviews (2)

excellent book on Fourier analysis
This is my all time favorite books on Fourier analysis for engineers. Weaver worked at Livermore Labs
and this book is based on a course for the technical staff given there. I have seen all the classics
Bracewell, Papoulis, Oppenheim, Goodman, etc... and I think this book blows them vis-a-vis developing
an elementary presentation of this material(Fourier Series, Fourier Transform, DFT) in a readable and concise way.To boot, the book also really starts at the beginning (i.e., what is an angle, what are trig functions, what is the derivative of sine and cosine...) Applications of the theory to areas such as optics and vibrations are also stressed.Great technical book!

"Applications..." yields amazing Clarity on Transforms
Mr. Weaver's "Applications ..." has saved me twice.Here's how:

I was warned by a graduating Electrical Engineering Senior to study Fourier Transforms before taking ECE-303, Signals and System theory.Quite simply, ECE-303 was the weeding course, because most students were struggling with the transforms while also trying to learn system theory.Fortunately, I heeded the senior's advice and picked up copy of "Applications ...".At the time, it was a bit much for my tight student budget, but it earned its keep.I read the book over the summer before the class.Because the book is so well written, I was able to digest it completely on my own.It has the greatest clarity I've seen on the topic.Having an Engineering disposition, I immediate consumed his explanations that tie things back to the "frequency content" view.I not only aced the class, but I really enjoyed it.It's what Engineering was about.

Here I am (fast forward 10 years) taking an Engineering Statistics class to get back into things.Low and behold, Weaver comes in again.In the advanced areas of studies in statistics I've picked up on references to the Fourier Transform in the topics of moments, moment generating functions, and the Central Limit Theorem.As it happens, Chapter 11, Stochastic Analyis covers this topic, and I've now got the concepts.Of course, I've had to re-read Chapter 3, The Fourier Transform, since it has been, after all, a great while since I applied them! ... Read more

Product Description

The book is written for graduate students who have read the first book and like to see the proofs which were not given there and/or want to see the full extent of the theory. On the other hand it can be read independently from the first one, only a modest knowledge on Fourier series/tranform is required to understand the examples.

This book fills a major gap in the textbook literature, as a full proof of Pontryagin Duality and Plancherel Theorem is hard to come by. It is usually given in books that focus on C*-algebras and thus carry too much technical overload for the reader who only wants these basic results of Harmonic Analysis. Other proofs use the structure theory which carries the reader away in a different direction. Here the authors consider the Banach-algebra approach more elegant and enlighting. They provide a streamlined approach that reaches the main results directly, and they also give the generalizations to the non-Abelian case.

Another main pillar of Harmonic analysis is the Poisson Summation Formula. We give its generalization to LCA-groups. The Selberg Trace Formula is considered the generalization of the Poisson Formula to non-abelian groups. The authors give the first textbook approach to this deep and useful formula in full generality. The last two chapters are devoted to examples of applications of the Selberg Trace Formula.

Product Description
Fourier analysis is one of the most useful and widely employed sets of tools for the engineer, the scientist, and the applied mathematician. As such, students and practitioners in these disciplines need a practical and mathematically solid introduction to its principles. They need straightforward verifications of its results and formulas, and they need clear indications of the limitations of those results and formulas.Principles of Fourier Analysis furnishes all this and more. It provides a comprehensive overview of the mathematical theory of Fourier analysis, including the development of Fourier series, "classical" Fourier transforms, generalized Fourier transforms and analysis, and the discrete theory. Much of the author's development is strikingly different from typical presentations. His approach to defining the classical Fourier transform results in a much cleaner, more coherent theory that leads naturally to a starting point for the generalized theory. He also introduces a new generalized theory based on the use of Gaussian test functions that yields an even more general -yet simpler -theory than usually presented.Principles of Fourier Analysis stimulates the appreciation and understanding of the fundamental concepts and serves both beginning students who have seen little or no Fourier analysis as well as the more advanced students who need a deeper understanding. Insightful, non-rigorous derivations motivate much of the material, and thought-provoking examples illustrate what can go wrong when formulas are misused. With clear, engaging exposition, readers develop the ability to intelligently handle the more sophisticated mathematics that Fourier analysis ultimately requires. ... Read more

Customer Reviews (2)

Principles of Fourier Analysis
I'm a junior student majoring in EE. This book is the best one on the market which provides a friendly approach on the subject for a beginner like me. I strongly recommend this book to any engineering or physics students. You will use this book over and over even in your major.
However, it would be even better if Dr. Howell can publish a software application book goes with this fantastic text.

Edward

I really Like this Book
This book is wonderful. It is quite clear and is written at a level for seniors and beginning graduate students.

It presentation is not as intense and terse as Lighthill's and Zemanian's on this subject. There is a good balance between the rigor and pedagogy that is quite appealing to me, along with examples and exercise with some answers/hints in the back for the end of chapter exercises.

Hopefully, Dr. Howell will update this book soon if needed. Also, it would be nice if the book were available in the size, shape and price of the books published by Dover. The book is quite big, thick and costs a hundred bucks.

Overall, I think many like/would like this book - it seems to get much use in the Library. ... Read more

Product Description
Real Analysis and Applications starts with a streamlined, but complete, approach to real analysis. It finishes with a wide variety of applications in Fourier series and the calculus of variations, including minimal surfaces, physics, economics, Riemannian geometry, and general relativity. The basic theory includes all the standard topics: limits of sequences, topology, compactness, the Cantor set and fractals, calculus with the Riemann integral, a chapter on the Lebesgue theory, sequences of functions, infinite series, and the exponential and Gamma functions. The applications conclude with a computation of the relativistic precession of Mercury's orbit, which Einstein called "convincing proof of the correctness of the theory [of General Relativity]." The text not only provides clear, logical proofs, but also shows the student how to derive them. The excellent exercises come with select solutions in the back. This is a text that makes it possible to do the full theory and significant applications in one semester. Frank Morgan is the author of six books and over one hundred articles on mathematics. He is an inaugural recipient of the Mathematical Association of America's national Haimo award for excellence in teaching. With this applied version of his Real Analysis text, Morgan brings his famous direct style to the growing numbers of potential mathematics majors who want to see applications along with the theory. The book is suitable for undergraduates interested in real analysis. ... Read more

Customer Reviews (1)

Great Intro to Advanced Math
" Real Analysis and Applications" by Frank Morgan is probably one of the best introductory analysis books that I have come across. It gives a basic overview over real analysis without getting too hung up on a specific topic. Though it includes most important proofs, it stays away from too complex and lengthy ones. The book contains a very detailed explanation that goes with every proof. The author keeps complicated matter as simple as possible and approaches it somewhat with humor. It is a great introductory book, especially for people that haven't been exposed to the material before. If you are already an analysis professional you probably might want to stay away from this book, since this book is really primarily for the advanced calculus beginner. ... Read more

Product Description
Fourier Methods in Imaging introduces the mathematical tools for modeling linear imaging systems to predict the action of the system or for solving for the input. The chapters are grouped into five sections, the first introduces the imaging “tasks” (direct, inverse, and system analysis), the basic concepts of linear algebra for vectors and functions, including complex-valued vectors, and inner products of vectors and functions. The second section defines "special" functions, mathematical operations, and transformations that are useful for describing imaging systems. Among these are the Fourier transforms of 1-D and 2-D function, and the Hankel and Radon transforms. This section also considers approximations of the Fourier transform. The third and fourth sections examine the discrete Fourier transform and the description of imaging systems as linear "filters", including the inverse, matched, Wiener and Wiener-Helstrom filters. The final section examines applications of linear system models to optical imaging systems, including holography.

• Provides a unified mathematical description of imaging systems.
• Develops a consistent mathematical formalism for characterizing imaging systems.
• Helps the reader develop an intuitive grasp of the most common mathematical methods, useful for describing the action of general linear systems on signals of one or more spatial dimensions.
• Offers parallel descriptions of continuous and discrete cases.
• Includes many graphical and pictorial examples to illustrate the concepts.

This book helps students develop an understanding of mathematical tools for describing general one- and two-dimensional linear imaging systems, and will also serve as a reference for engineers and scientists

Product Description
Significantly revised and expanded, this new Second Edition provides readers at all levels---from beginning students to practicing analysts---with the basic concepts and standard tools necessary to solve problems of analysis, and how to apply these concepts to research in a variety of areas.

Authors Elliott Lieb and Michael Loss take you quickly from basic topics to methods that work successfully in mathematics and its applications. While omitting many usual typical textbook topics, Analysis includes all necessary definitions, proofs, explanations, examples, and exercises to bring the reader to an advanced level of understanding with a minimum of fuss, and, at the same time, doing so in a rigorous and pedagogical way. Many topics that are useful and important, but usually left to advanced monographs, are presented in Analysis, and these give the beginner a sense that the subject is alive and growing.

This new Second Edition incorporates numerous changes since the publication of the original 1997 edition and includes:

Features:

a new chapter on eigenvalues that covers the min-max principle, semi-classical approximation, coherent states, Lieb-Thirring inequalities, and more

extensive additions to chapters covering Sobolev Inequalities, including the Nash and Log Sobolev inequalities

new material on Measure and Integration

many new exercises

and much more ...

The Second Edition continues its no-nonsense approach to the topic that has made it one of the best selling books on the subject. It is an authoritative, straight-forward volume that readers---from the graduate student, to the professional mathematician, to the physicist or engineer using analytical methods---will find useful both as a reference and as a guide to real problem solving.

About the authors: Elliott Lieb is Professor of Mathematics and Physics at Princeton University and is a member of the US, Austrian, and Danish Academies of Science. He is also the recipient of several prizes including the 1988 AMS/SIAM Birkhoff Prize. Michael Loss is Professor of Mathematics at the Georgia Institute of Technology. ... Read more

Customer Reviews (3)

A tremendous jumpstart into modern mathematical physics
I am a physicist with a somewhat limited mathematical background. However, I decided to 'break in' in modern mathematical physics, and that meant acquiring first a modicum of functional analysis and the required measure theory, harmonic analysis, and operator theory that goes with it. If you can afford the time, the classical learning path through, e.g., Kolmogorov & Fomin > Rudin (R&C) > Reed & Simon I (a path that I recommend, by the way), with possible excursions into ODEs and PDEs, probability theory, and modern geometry, is the safe way to go. If you cannot afford the time, read Lieb & Loss. It provides a tremendous jump start into what really matters for a beginning mathematical physicist: a little measure theory, L^p spaces, Sobolev spaces, a bit of Fourier analysis and PDEs, and inequalities -- lots of them (integral, Sobolev, variational). The main theorems are stated, most with proofs, and put into use.

I had a mathematical physics teacher in graduate school that once said (somewhat half-jokingly) that all you need to know is the monotone and dominated convergence theorems, the Fubini theorem, the Borel-Cantelli lemma, the Euler-Lagrange equations and how to resolve the identity using plane waves. This book by Lieb & Loss is a testimony to his confession.

A start in analysis.
A start in analysis.-- For some number of years, Rudin's "Real and Complex", and a few other analysis books, served as the canonical choice for the book to use, and to teach from, in a first year grad analysis course. Lieb-Loss offers a refreshing alternative: It begins with a down-to-earth intro to measure theory, L^p and all that...It aims at a wide range of essential applications, such as the Fourier transform, and series, inequalities, distributions, and Sobolev spaces,--- PDE, potential theory, calculus of variations, and math physics (Schrodinger's equation, the hydrogen atom, Thomas-Fermi theory... to mention a few.) The book should work equallly well in a one, or in a two semester course. The first half of the book covers the basics, and the rest will be great for students to have, regardless of whether or not it gets to be included in a course. This choice of book is also especially agreeable to grad students in physics who need to read up on the tools of analysis.

An excellent course of analysis with a theme
By the end of the sixties Dyson and Lennard, for the first time, proved that matter is stable. More precisely, they proved the thermodynamic stability of Coulomb matter. This was a landmark of mathematical physics, and a huge one: a very long and hard paper. A few years later, Elliott Lieb and Walter Thirring substantially improved the great Dyson result, dramatically cutting its length while improving important estimates. A very good review of these results can be find in the volume 4 of Thirring's "A Course in Mathematical Physics". Even the book version is a bit hard to read, as much mathematical analysis is required. The "Analysis" of Lieb and Loss is a book on analysis which has as a theme the great result of Lieb and Thirring. It is a real book on analysis. The chapters are named "Measure and Integration", "Lp-spaces","The Fourier transform", "Distributions", but also "Potential Theory and Coulomb En! ergies" and "Introduction to the Calculus of Variations", where nothing less than the Thomas-Fermi atom is rigorously studied. In order to leave no doubt that hard analysis is present, there are two chapters on Inequalities. After studying this splendid text the reader will be a better analist and, if he cares to, can start reading the proof of stability of matter. The proof of the pudding is NOT in the eating! ... Read more

Product Description

A carefully prepared account of the basic ideas in Fourier analysis and its applications to the study of partial differential equations. The author succeeds to make his exposition accessible to readers with a limited background, for example, those not acquainted with the Lebesgue integral. Readers should be familiar with calculus, linear algebra, and complex numbers. At the same time, the author has managed to include discussions of more advanced topics such as the Gibbs phenomenon, distributions, Sturm-Liouville theory, Cesaro summability and multi-dimensional Fourier analysis, topics which one usually does not find in books at this level. A variety of worked examples and exercises will help the readers to apply their newly acquired knowledge.

Customer Reviews (1)

Olle R
I have waited for some professional mathematician to review this book, since I think it is such a good book. This has not happened, so here is my un-professional review:
I am a retired electrical engineer, and the last three years I have worked on refreshing my old and forgotten school math, and also to going a bit further than during school. When I bought this book I had the following wishes, all of whom it fulfilled brilliantly:
1. It should be usable for self study.
2. It should refresh my fourier series and transforms.
3. It should put these things in context, by presenting physics problems and also by widening the scope to wet the appetite for functional analysis and partial diff eqs.
4. It should do something for my sorely lacking "mathematical maturity".
All of these things it did and I just want to add that Mr. Vretblad is an outstanding pedagogue and that the book contains very few typos. (I found two or three in the whole book, none of them important).
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Product Description
Text provides an exposition of the basic ideas of the theory of distribution and Fourier transforms and its application to partial differential equations. Introduces several recent topics, including pseudodifferential operators, wave front sets, wavelets, and quasicrystals. For advanced undergraduate and graduate students. Previous edition: c1994. Softcover, hardcover available. ... Read more

Customer Reviews (1)

Excellent
Distributions are objects most physicists will frequently encounter during their career, but, surprinsingly, the subject is not given the place it deserves in the current ordinary science curriculum.
I would particularly recommend this book to physics students willing to learn the foundation of distribution theory and its close ties to Fourier transforms. Distribution theory is, basically speaking, a way of making rigorous the operations physicists find Ok to carry on functions, that otherwise wouldn't rigorously make sense. Distribution theory therefore provides a useful way of checking, in the process of a calculation, if it is allowed (according to the extended rules of distribution theory), or if it is definitely dubious (e.g. current distribution theory doesn't provide a mean of making sense of a product of Dirac delta functions, while such expressions sometimes pop out in the context of quantum field theory ; nevertheless, there exist other formal theories, such as Colombo calculus that aim at justifying this ; yet, for some reason, they seem to bear less power than the original distribution theory).
This work is an easy, gentle, pedagogical piece of mathematical exposition.
The subject is wonderfully motivated.
As such, this book is suited to self-study.
It could also be used as a textbook for an introductory course on the subject, or as an introductory reading to more advanced texts (Aizenman, for instance).
Highly recommended. ... Read more

Product Description
Discover applications of Fourier analysis on finite non-Abelian groups

The majority of publications in spectral techniques consider Fourier transform on Abelian groups. However, non-Abelian groups provide notable advantages in efficient implementations of spectral methods.

Fourier Analysis on Finite Groups with Applications in Signal Processing and System Design examines aspects of Fourier analysis on finite non-Abelian groups and discusses different methods used to determine compact representations for discrete functions providing for their efficient realizations and related applications. Switching functions are included as an example of discrete functions in engineering practice. Additionally, consideration is given to the polynomial expressions and decision diagrams defined in terms of Fourier transform on finite non-Abelian groups.

A solid foundation of this complex topic is provided by beginning with a review of signals and their mathematical models and Fourier analysis. Next, the book examines recent achievements and discoveries in:

• Matrix interpretation of the fast Fourier transform
• Optimization of decision diagrams
• Functional expressions on quaternion groups
• Gibbs derivatives on finite groups
• Linear systems on finite non-Abelian groups
• Hilbert transform on finite groups

Among the highlights is an in-depth coverage of applications of abstract harmonic analysis on finite non-Abelian groups in compact representations of discrete functions and related tasks in signal processing and system design, including logic design. All chapters are self-contained, each with a list of references to facilitate the development of specialized courses or self-study.

With nearly 100 illustrative figures and fifty tables, this is an excellent textbook for graduate-level students and researchers in signal processing, logic design, and system theory-as well as the more general topics of computer science and applied mathematics. ... Read more

Product Description
Introducing the Fourier Transform as a useful and pratical tool in the biomedical sciences, this volume provides the reader with practical demonstrations and examples. ... Read more