Book Description

Wavelets are a mathematical development that may revolutionize the world of information storage and retrieval according to many experts. They are a fairly simple mathematical tool now being applied to the compression of data--such as fingerprints, weather satellite photographs, and medical x-rays--that were previously thought to be impossible to condense without losing crucial details.

This monograph contains 10 lectures presented by Dr. Daubechies as the principal speaker at the 1990 CBMS-NSF Conference on Wavelets and Applications. The author has worked on several aspects of the wavelet transform and has developed a collection of wavelets that are remarkably efficient.

The opening chapter provides an overview of the main problems presented in the book. Following chapters discuss the theoretical and practical aspects of wavelet theory, including wavelet transforms, orthonormal bases of wavelets, and characterization of functional spaces by means of wavelets. The last chapter presents several topics under active research, as multidimensional wavelets, wavelet packet bases, and a construction of wavelets tailored to decompose functions defined in a finite interval. Because of their interdisciplinary origins, wavelets appeal to scientists and engineers of many different backgrounds. ... Read more

Reviews (6)

Over rated
There are so many well written books on Wavelts out nowadays. Don't waste your money on this one. It's famous because it was first (or one of the first). On the positive side, it does present a strong mathematical foundation. I recommend you buy a readable book (just do a search on Amazon.com and you'll find half a dozen great books on Wavelets), then when you approach the "expert" level, use this one a s a reference (if at all).

Great for everyone!
This book has become a classic,-- and a hit;-- for more than ten reasons. It is multilayered, and yet presents a unity of ideas: The material, and the writing is engaging for the beginner, and for the research mathematician alike. When I used it in my teaching, it was equally popular with the math students, and those from engineering. I don't know if I can say this about any other book I have taught from. The students could follow all the carefully presented proofs, and the engineer could generate algorithms from the applied chapters.

Not for the faint of heart!
This book is a treasure of details if you know what you are doing. As another reviewer noted, it is not for the beginner. But if you have had some experience with the subject this is a must have for your library shelf.

Theoretical Wonder....but you won't learn wavelets from it!
This is the document that started it all. It is by far a great mathematical and theoritical piece of work. HOWEVER, if you are just starting off and want to learn about wavelets and do not have an advanced math or engineering degree (and I do mean ADVANCED), do not pick up this book. At least not at the beginning. There are much better books written for explaining wavelets and to better present the material. Ten Lectures is essentially one big proof. Try Mallat/Kovacevic or Strang...once you've got a solid understanding, come back to Daubechies and marvel at her work.

Ten Lectures on Wavelets
This book is classic. For anyone who wants really to know wavelets, in particular comapctly supported wavelets, this book is a must-read. Also, this book is truly well-written, which makes it suitable for text of a beginning gradaute course. Thanks to Dr. Dabechies! ... Read more

Book Description

This book fills the gap between volumes on wavelets that are either too advanced (in terms of mathematical background required) or that contain too little mathematical theory underlying wavelets.It presents most of the theory underlying Fourier analysis and wavelets in a clear and comprehensive fashion-- without requiring advanced background in real analysis. Provides a careful balance between theory and practical algorithms, and features a clear presentation of applications to digital signal processing--e.g., data compression, digital filtering and singularity detection. Includes illustrations and MATLAB code used in many of the algorithms.Inner Product Spaces. Fourier Series. The Fourier Transform. Discrete Fourier Analysis. Wavelet Analysis. Multiresolution Analysis. The Daubechies Wavelets.For anyone interested in Wavelets and Fourier Analysis. ... Read more

Reviews (2)

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.

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 ordinary differential equations, 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 of chapter 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 with wavelets 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

Book Description

Written by a successful author and respected mathematician, this book emphasizes a concrete and computational approach to the subject of Fourier analysis and wavelet theory while maintaining a balance between theory and applications. In some cases, several different proofs are offered for a given proposition, allowing students to compare different methods. ... Read more

Reviews (1)

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

Book Description

Published by McGraw-Hill since its first edition in 1941, this classic text is an introduction to Fourier series and their applications to boundary value problems in partial differential equations of engineering and physics.It will primarily be used by students with a background in ordinary differential equations and advanced calculus.There are two main objectives of this text.The first is to introduce the concept of orthogonal sets of functions and representations of arbitrary functions in series of functions from such sets.The second is a clear presentation of the classical method of separation of variables used in solving boundary value problems with the aid of those representations. ... Read more

Reviews (4)

Excellent
This book is quite thorough, but remains easy to follow (considering the material). It starts out with partial differential equations (no previous PDE experience needed) and shows where Fourier series comes from, which I found motivating since the purpose of Fourier Analysis was evident from the beginning. It then goes into making solutions of arbitrary functions out of sine and cosine functions as well as touching on other orthogonal sets.

The book's main focus is on starting with PDEs and ending with a solution of a Fourier series.

The first chapter was the hardest since the approaches to problems were much different than in calculus, but after adjusting to the material and the approaches to the problems, it gets easier!

Great text for an intro to pde's course!
My first encounter with partial differential equations was out of this book. Since then, I've had another course on pde's, and used this book as a reference quite often. Fourier Series adn Boundary Value Problems is very much like Complex Variables and Applicatoins, also by Churchill and Brown. It's accessible to a large audience. Though it would help to have had an advanced calculus course, it isn't necessary to understand the mechanics of solving pde's (namely the variables seperable cases, which is mostly what's in this book). If you're an undergraduate math, engineering or physics student, you'll probably be using this book.

Try Another Text
I found Dr. Brown, in conjunction with Dr. Churchill, to have written a very dry and non-useful text. It fails to provide the undergraduate student with the resources and background information that more highly touted books offer. There are a few examples that are somewhat helpful, but overall I found myself having to use reference texts to supplement this one. I am not a math major, but am continually searching for good math texts to help me grasp the fundamentals of more difficult topics. I did not find that help here. Too much 'math prose' and not enough to-the-point definitions and examples, which is the cry of every non-math major. Their treatment of the Laplacian is not even worth the bother of placing it in the book. The physical size of the book is small, (9 1/2 by 6") with 335 pages. Not nearly enough for the treatment of its titled subject.

An excellent book on Fourier Series
This is a great book that gives precise examples which are easy to comprehend. Dr. Brown proves to be an excellent author once again. ... Read more

Book Description

Reviews (2)

I recommend this book to anyone who loves mathematics.
A Y Khinchin was one of the greatest mathematicians of the first half of the twentieth century. His name is is already well-known to students of probability theory along with A N Kolmogorov and others from the host of important theorems, inequalites, constants named after them. He was also famous as a teacher and communicator. Several of the books he wrote are still in print in English translations, published by Dover. Like William Feller and Richard Feynman he combines a complete mastery of his subject with an ability to explain clearly without sacrificing mathematical rigour.

In this short book the first two chapters contain a very clear development of the theory of simple continued fractions, culminating in a proof of Lagrange's theorem on the periodicity of the continued fraction representation of quadratic surds. Chapter three presents Khinchins beautiful and original work on the measure theory of continued fractions. The proofs of the theorems in this chapter are also entirely elementary.

Classic text, however not suitable for a first exposure.
This is Khinchin's classic work, translated from Russian in the 1930's. Although the book is rich with insight and information, Khinchin stays one nautical mile ahead of the reader at all times, the book moves at a truly alarming pace, and the book is not suitable to be used ALONE as an introduction to continued fractions. To supplement this book if this is a first exposure to continued fractions, I would recommend C.D. Old's book, which has many more examples which can be worked through until the reader is comfortable with the topic.

The book is brilliant and necessary for understanding continued fractions, but can't stand alone without supplemental material unless one is a professional mathematician. Khinchin frequently employs contrapositive proof formats, and there are occasional translation errors from Russian. The errors range from minor (awkward usage) to major (in one place, the translation is "negative" when it should be "non-negative", which confused me for half a day). ... Read more

Book Description

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

Reviews (2)

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.

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

Book Description

Awarded the American Mathematical Society Steele Prize for Mathematical Exposition, this Introduction, first published in 1968, has firmly established itself as a classic text. Yitzhak Katznelson demonstrates the central ideas of harmonic analysis and provides a stock of examples to foster a clear understanding of the theory. This new edition has been revised to include several new sections and a new appendix. ... Read more

Reviews (3)

Still one of the best.
When the first edition of Katznelson's book appeared back in 1968 (when I was a student), it soon became the talked about, and universally used, reference volume for the standard tools of harmonic analysis: Fourier series, Fourier transforms, Fourier analysis/synthesis, the math of time-frequency filtering, causality ideas, H^p-spaces, and the various incarnations of Norbert Wiener's ideas on the Fourier transform in the complex domain, Paley-Wiener, spectral theory, and more. It is easy to pick up the essentials in this lovely book. Now, many years later, I occasionaly ask beginning students what their favorite reference is on things like that, and more often than not, it is Katznelson. Thanks to Dover, it is on the shelf of most university bookstores, and priced under US$ 10.

Great Introduction to Classical Harmonic Analysis
This is a great book for looking at classical harmonic analysis: the study of Fourier Series on the "typical" groups, includes a quick look at the general situation and ends with an introduction to commutative Banach Algebras. Both topics are continued in [Loomis].

An essential book for anyone studying Harmonic Analysis
Katznelson's book considers harmonic analysis primarily on the circle group. He does this from a thorougly modern point of view. An understanding of the basic ideas of Banach spaces is required. This book should be on the shelf of any aspiring Harmonic Analyst, especially one with an abstract viewpoint. ... Read more

Reviews (1)

Review
John D. Blanton did a good job in translating Euler's work on analysis of the infinite. Although I did not read the original copy the book, I could still receive the charm of Euler from the translation. Euler wrote this book because he realized that many mathematics students were lacking of the knowledge of analysis of the infinite. He, as a great master of mathematics and educator and with all his passion, presented this timeless masterpiece to all of us. I recommend this book to every students who want to excel in mathematics! ... Read more

Amazon.com

Before discussing the merits of David Foster Wallace's Everything and More: A Compact History of Infinity, it is essential to define what the book is not. This volume in the "Great Discoveries" series is not a history of the personalities and social conditions that led to the "discovery" of infinity. Nor is it a narrative fixated on the cultish fear of--and obsession with--the infinite that has seemingly driven mathematicians insane over the centuries. Rather, Everything and More is a surprisingly rigorous march through the 2000 plus years of mathematical research that began with Aristotle; continued through Galileo, Isaac Newton, G.W. Leibniz, Karl Weierstrass, and J.W.R. Dedekind; and culminated in Georg Cantor and his Set Theory.The task Wallace (author of the bestseller Infinite Jest and other fiction) has set himself is enormously challenging: without radically compromising the complexity of the philosophy, metaphysics, or mathematics that underlies the evolving concept of infinity, present the material to a lay audience in a manner that is entertaining. To propel his narrative, Wallace even develops a style that mirrors the mathematical language he probes. One difficulty in his focus on concepts and not a strict human chronology, though, is that his structure is dependent on frequent digressions (especially early on). Patience is required. Wallace demands that his reader walk through the equations, study the graphs and charts, and relearn college-level concepts to follow along on the exploration. Indeed, after one wrenching dip into Zeno’s paradoxes, Wallace spouts at his imagined complaining audience: "Deal." But the book should be deemed a success. If one grants him the attention he requires, Wallace has made the trip richly rewarding.--Patrick O’Kelley ... Read more

Reviews (32)

aleph alpha nought nought one
i've been pretty excited to get my hands on this for some time now, so admittedly my expectations were pretty high. the book feels in many ways as if it were a patch-work job. excerpts of writing blaze (like screaming, but for the eyes) with wallace's style, yet other bits want for lucidity and are, rather surprisingly, not exactly enjoyable. for example, there are various non-independent-clause/sentances. this i expect (and find fitting) from delillo, but wallace doesn't seem to make very good use of such a construct (and generally it is cause for a rather harsh break in any fluidity built up to said clause-as-sentence).

now it's probably of some import here that i'm a graduate mathematics student and was really just hoping to read one of my favourite writers thoughts on a subject that i enjoy, but mathematically this book is less-than-ideal. i don't know if someone without the "college math" he so often refers to will be able to get much from his explication of the various mathematical ideas he is presenting (even some of the very early examples meant to ellucidate the paradoxical situations that arise when dealing with infinity as a cardinal are (unnecessarily) confusing).

so, basically, read the book. it's wallace at times, and those times make it worthwhile. if you want an introduction to set theory, look elsewhere (even to cantor himself), and then come back and read this because it really is a nice book at times (i mean (tautologically), when he's on, he's on).

p.s. something i'd meant to mention the first time around: wallace discusses some (of the many) ways in which infinity gives us trouble, and he speaks (often at length) about various interesting aspects of these difficulties, but he fails entirely to mention a most important fact: we have no "direct" word for the infinite. our only means of describing these objects is to call them non-finite. this linguistic/conceptual failing occurs not once, but twice, in that we have various infinities of two basic types: countable (the "smaller" of the two) and (you guessed it) uncountable. that he failed to cover this is, i think, quite representative of the failings of this book. but again, i highly recommend the book, 100% (ummmm, you see, "100%" is one of the shining moments of this book, but until you've read it, you won't really get to enjoy that. a shame, no?)

Good, but flawed
Good subject, interesting (although sometimes tangled) presentation.

But the mathematical mistakes just spoil everything. Like the proof of dichotomy convergence using Weierstrass delta-epsilon thing for continuity. What was that? Looked like the author himself didn't quite understand what he was trying to do, so he just crumpled the proof: "Hence... Hence...".

A fellow of infinite jest?
I found this a well-researched book by a knowledgeable author, marred only by his putting it into words. The presentation is perhaps that of an overdue term paper at three a.m., when the un-numbered pages have collapsed into a pile on the floor.

The arrogant mannerisms, cliches and hackneyed phrases, ideosyncratic abbreviations, and lack of linear structure make it a book that, once you put it down, is hard to pick up again.

I bought this book hoping to bring away from it some fresh perspectives on infinity, to benefit the calculus students I am teaching. I left it empty-handed.

Paradoxically flawed
Inspired by praise for David Foster Wallace's "Everything and More" in publications including The Onion and Wired, I bought it hoping to revive in myself and instill in my kids an enduring excitement about mathematics.

Wallace begins with a series of anecdotes that promised to fill the bill, leavened with plain talk and a bracing occasional bit of scatology. But the book's reliance on advanced notation -- much of it impenetrable even to this reader, despite four years of college math (up to differential equations!) -- soon kills the narrative flow.

Wallace's parenthetical asides and copious footnotes sometimes provide illumination, but the book's scattershot structure belies the dust jacket's promise of "a literary masterpiece."

Even Wallace himself acknowledges the book's shortcomings, apologizing at several points for convoluted sentences, bewildering explanations and jumbled storytelling. A good editor could have helped him cut those knots, isolating the advanced math or otherwise rendering it intelligible, allowing him to deliver what author James Gleick hails in his promotional blurb as "exquisitely (and hilariously) original science writing." (Did Gleick and the other reviewers survive the entire book? Or did they just get the funny parts?)

Reading "Everything and More" was like being trapped in a literary version of Zeno's Paradox: Finishing half the book, then struggling to complete half of what remained, then half of that ... I finally just gave up, disillusioned.

Everything and More: A Compact History of Infinity
Wallace's writing about math isn't new-his novel Infinite Jest (1996) and some of his essays include a more than superficial treatment of the subject. Here, however, he digs as deeply into it as is possible for a nonprofessional math geek faced with a page limit, and the result is classic DFW: engaging, self-conscious, playful, and often breathless. This second installment in the "Great Discoveries" series traces the history of infinity from the Greeks to the calculus, culminating in a discussion of Georg Cantor's (1845-1918) groundbreaking work with transfinite numbers. Unfortunately, context requires Wallace to bulldoze heroically through a couple thousand years of logic, geometry, and number theory, which, even with "emergency glossaries" and frequent cross-referencing tips, can make for some teeth-grindingly dense passages. In one of the 400-plus footnotes, he writes, "It's true that it would be nice if you've had some college math, but please rest assured that considerable pains have been taken and infelicities permitted to make sure it's not required." For devout Wallace fans, it won't matter either way. Readers looking to soak up some pure abstraction, however, may just need to read the book twice. Luckily, they couldn't have been blessed with a more talented or stimulating guide. Enthusiastically recommended for all libraries. ... Read more

Reviews (1)

Important Contribution to Modern Epistemology
The 20th century saw more advances in knowledge than could filter down to general society. Relativity and Quantum Theory are part of the vernacular, even if the popular conceptions are not necessarily good generalizations of their counterparts in science. The corresponding advances in philosophy, however, have stayed more in the province of academia, largely because philosophy itself has become highly technical; but the physics of beyond-everyday-experience have demanded these advances, primarily in epistemology, because the fundamental questions of science today are of meaning and understanding.

Understanding the Infinite is a work of epistemology. Its contribution to the foundations of general knowledge demand that it disseminate beyond academia, although the ground Lavine breaks requires the extensive citations and technical style he employs. The author poses and addresses the following question. If set theory is so intuitively self-evident and seemingly such a fundamental underpinning of all mathematics, why is it so hard to express technically and why has the axiomatization of set theory been so controversial? Set theory was the big idea which the mid-20th century educational establishment thought important enough to indoctrinate schoolchildren with in the guise of new math. Yet set theory never took root in popular consciousness, certainly not the notion of transfiniteness.

Lavine starts out by dispelling the anecdotal account of the development of set theory, which has misled even professional mathematicians and philosophers to conclude "The fundamental axioms of mathematics...are to a large extent arbitrary and historically determined." He constructs what he claims is the correct historical development of set theory (I'll let historians of mathematics decide this) including sidetracks into Russell's failed program to equate mathematics and logic (and in the process dispels the significance of Russell's paradox), and von Neumann's axiomatization of set theory emphasizing functions. The outcome of his exposition is the Zermelo-Fraenkel axiomatization with the Axiom of Choice (ZFC), today's common form of set theory. These chapters by themselves could serve as an introduction to set theory, except that the Continuum Hypothesis is barely mentioned, since it plays no role in Lavine's program. Admittedly, he has nothing new to add.

The main event is Lavine's epistemological tour-de-force. Building upon work of Jan Mycielski he introduces the reader to the concept of finitary mathematics and constructs a finitary ZFC, showing that this theory justifies the adoption of what he calls the "Axiom of Zillions" (indefinitely large sets) in which we have access to very large sets' ordinal, but not necessarily its predecessors. The final step is to show this all "intuitively" extrapolates to ZFC.

QEF, QED.

I introduced physics in the opening paragraph of this review because I see Lavine's rigorous treatise in the epistemology of mathematics as a contribution to the grand unification of physics, mathematics, and epistemology. Lavine treads lightly in the physical realm. He writes "...modern physics makes it seem likely that the physical universe is of finite extent..." All of the dominant cosmologies put forth in the 20th century incorporated this misdirection set off by general relativity. On a large scale the universe must be curved. Ironically Lavine published in 1994, just as new astronomical observations began whispering "in three dimensions the universe is Euclidean". If that whisper becomes a shout in the 21st century, as appears likely from the mounting evidence, physics will have to address the transfinite.

The Calculus had to be put on a firm theoretical foundation so that it could be used as a tool to advance knowledge without justifying its use. We may see that Lavine's epistemology will do the same for set theory and transfinite numbers. ... Read more

Book Description

This companion volume to Andy Zonst's Understanding the FFT is written in five parts, covering a range of topics from transient circuit analysis to two dimensional transforms.It's an introducton to some of the many applicatons of the FFT, and it's intended for anyone who wants to understand and explore this technology.

The presentation is unique in that it avoids the calculus almost (but not quite) completely.It's a practical "how-to" book, but it also provides down to earth understanding.

This book developes computer programs in BASIC and the reader is encouraged to type these into a computer and run them; however, for those who don't have access to a BASIC compiler you may down load the programs from the internet (contact Citrus Press for URL).

The potential buyer should understand that presentations are frequently started at an elementary level.This is just a technique to establish the foundation for the subsequent discussion, intended for those who don't already understand the subject (the material usually comes quickly to the problem at hand).The book is written in an informal, tutorial style, and should be managable by anyone with a solid background in high school algebra, trigonometry, and complex arithmetic.Zonst has included the mathematics that might not be available in a high-school curriculum; so, if you managed to work your way through the first book, you should be able to handle this one.

For those familiar with the first edition of this book, the most prominant feature of this revised edition will be its improved coherence and readability. ... Read more

Book Description

Contains 36 lectures solely on Fourier analysis and the FFT. Time and frequency domains, representation of waveforms in terms of complex exponentials and sinusoids, convolution, impulse response and the frequency transfer function, modulation and demodulation are among the topics covered. The text is linked to a complete FFT system on the accompanying disk where almost all of the exercises can be either carried out or verified. End-of-chapter exercises have been carefully constructed to serve as a development and consolidation of concepts discussed in the text. ... Read more

Reviews (1)

Good choice for the novice interested in Fourier Analysis

Throughout my graduate career, I searched desperately for a book that covered Fourier Analysis in a manner that would be lucid to a novice. Of all the books that I've seen dealing with the subject, I rank Morrison's as the best in this category.

Most books dealing with Fourier Analysis appear to be written for someone with a degree in mathematics. Although I would say that this book is an excellent introduction to Fourier analysis and the Fourier transform, the reader must possess a strong working knowledge of calculus at the least.

The book is presented in the classic textbook format, where each section is introduced and explained with examples, then a series of problems are presented to reinforce the concepts presented.

The first half of the book covers continuous Fourier analysis, and the second half of the book covers discreet Fourier analysis. Some may argue that these two concepts could have been introduced simultaneously, however I found this dichomtomy to be an effective way of presenting the material. The book is geared towards undergraduate students of electrical engineering, but I think that it is appropriate for anyone wishing to learn Fourier analysis. The book is replete with exercises to be completed with the accompanying diskettes (both Mac and PC are included), but I never used them.

This book is the best I've seen dealing with the subject, but I did have to proceed very slowly. I did not understand all of the concepts presented, perhaps because of my limited mathematics background (as high as calculus). For this reason, I rate the book a seven. ... Read more

Book Description

Robert Kaplan's The Nothing That Is: A Natural History of Zero was an international best-seller, translated into eight languages. The Times called it 'elegant, discursive, and littered with quotes and allusions from Aquinas via Gershwin to Woolf' and The Philadelphia Inquirer praised it as 'absolutely scintillating.' In this delightful new book, Robert Kaplan, writing together with his wife Ellen Kaplan, once again takes us on a witty, literate, and accessible tour of the world of mathematics. Where The Nothing That Is looked at math through the lens of zero, The Art of the Infinite takes infinity, in its countless guises, as a touchstone for understanding mathematical thinking. Tracing a path from Pythagoras, whose great Theorem led inexorably to a discovery that his followers tried in vain to keep secret (the existence of irrational numbers); through Descartes and Leibniz; to the brilliant, haunted Georg Cantor, who proved that infinity can come in different sizes, the Kaplans show how the attempt to grasp the ungraspable embodies the essence of mathematics. The Kaplans guide us through the 'Republic of Numbers,' where we meet both its upstanding citizens and more shadowy dwellers; and we travel across the plane of geometry into the unlikely realm where parallel lines meet. Along the way, deft character studies of great mathematicians (and equally colorful lesser ones) illustrate the opposed yet intertwined modes of mathematical thinking: the intutionist notion that we discover mathematical truth as it exists, and the formalist belief that math is true because we invent consistent rules for it. 'Less than All,' wrote William Blake, 'cannot satisfy Man.' The Art of the Infinite shows us some of the ways that Man has grappled with All, and reveals mathematics as one of the most exhilarating expressions of the human imagination. ... Read more

Reviews (6)

As complex as math can be.
As if math wasn't complex and confusing enough, a book with equally confusing english was written about it. With out bragging, I am fluent in mathmatics; I understand it as if it were my primary language. What I am not fluent in is English, and unfortunately this book was written only for the English elite. 1/4th of the time I understood half of the poetic correlations between mathmatics and philosophy described in this book, which, consequently happens to be 3/4ths of the context. Basically, if you understand mathematics as well as I do, but do not understand poetry and philosiphy well, do not read this book, you're well off where you are. If you do understand English, extreemly well, and want to know more about mathematics, read the book. But if you could care less about mathematics, or english, then don't even read this review.

Is the Prose Delightfully or Excessively Rich?
As you can read from other reviews, this book rates 5 stars for its excellent description and illustration of many fascinating topics in mathematics. Not all readers, in contrast, will appreciate the authors' most unusual prose style. At times they can't seem to write a sentence without a metaphor, and often a startling or even madcap one. Allusions, philosophical insights, snatches of poetry and unusual quotations, verbs that wriggle or hop--they are all crammed together. So at times the mathematics seems a good deal easier to handle than the prose.

I was at first tempted just to dismiss this style as mere overwriting, but as I read further I started to see that it nicely fit the remarkable turns of thoughts of the master mathematicians as they tested their brains on the challenges of number and space. The more-than-quirky prose, including its philosophical and quasi-religious asides, definitely adds to the interest and instructiveness of the book, I finally decided.

This book is, as you can imagine, far more absorbing than the school math most of us were subjected to. Five stars.

The proof of a.0 = 0 is incomplete.
In the proof on page 40 of a.0 = 0,

Line 1: a.0 = a(1-1)
Line 2:.......= a - a
Line 3:.......= 0

since (1-1) is shorthand for 1+(-1), distributivity only yields

a(1-1) = a[1+(-1)] = a.1+a(-1)

so that going from Line 1 to Line 2 implicitly assumes that a(-1) is equal to -a, which has not been previously established from the axioms.

Great Math..... Obscure Prose
The mathematics in this book is clear and absolutely delightful - reminiscent of high school math. The derivations, proofs, figures and equations are all very clear and the words immediately associated with them are very useful complements. The problem arises when we are in-between the mathematical expositions, i.e., where historical and miscellaneous other snipets are presented; these would normally be pleasant diversions and would make the book even more interesting. But here, this is not the case. The prose is rather obscure, complex and cryptic and tends towards the quasi-poetic, quasi-philosophical and quasi-parabolic all at once. This is most unfortunate for a math book where simplicity and clarity of expression are paramount. Had the historical and other digressions been written clearly and in plain English, then this book, in my opinion, would have easily been 5-star material. But as it is, the math is worth an easy 5 stars, the prose an unfortanate 1 star for an average of 3 stars.

To Infinity, And Beyond!
We all take our pleasures where we find them, and everyone is different, with different sources to draw upon. It will seem peculiar to many people that others could take pleasure in mathematics. Children usually learn to be bored or frightened by math, but there isn't any reason for this, other than incompetent teaching. As an attempt at remedy, husband and wife team Robert and Ellen Kaplan in 1994 began the Math Circle, Saturday morning sessions for kids who just wanted to find out more about mathematics. (The sessions were changed to Sunday morning when soccer practice interfered). Some kids (especially those who were pushed into the classes by their parents) dropped out, but some have come back, year after year, and the Kaplans have found that posing questions, inviting conjectures, asking for examples, and even suggesting ways towards proofs can be something children can enjoy. Mathematicians have been telling us for centuries about the beauty of the objects and systems that they have explored. The Math Circle seems to have taught math in a way to at least some kids who have caught the spirit of the quest for mathematical beauty. In _The Art of the Infinite: The Pleasures of Mathematics_ (Oxford University Press), the Kaplans have put some of those lessons into book form, concentrating on infinities of various kinds. This is a book for adults, or kids who hanker to think about math like adults ought to, but it is full of a sense of play.

As you might expect, things start simple and get very complicated, and this is true right off in the first chapter, considering more and more complicated numbers. The Natural Numbers are introduced with patterns, as if you had stones to position on a table. 1, 3, 6, and 10 stones make pleasing equilateral triangles, and 1, 4, 9, and 16 make pleasing squares. We move from these to zero and negative numbers: "Certainly zero and the negatives have all the marks of human artifice: deftness, ambiguity, understatement." Are these numbers invented or discovered? The profundity of this question is plumbed throughout the book. Rationals, irrationals, and finally the complex numbers are all included. As the numbers mount up, the irregularity and regularity of the primes is considered, one of the most fruitful arenas of number theory. Euclid had to make an assumption about the infinite, his famous fifth postulate; but it is only an assumption; assuming that parallel lines meet eventually produces also a worthy geometry that tells us much about how the Einsteinian universe works. But there is no need to look into these strange worlds to find wonders; before leaving Euclid's terra firma, we are reintroduced to the triangle, and are presented with some astonishing revelations of secret points within and around the simple three sides that will remind you that no matter how simple things look, or even how simple things are, everything is more complicated than you can imagine.

And if you want your infinities more complicated still, the final chapter has to do with Cantor's work. Common sense tells us there must be half as many even numbers as there are whole numbers, but Cantor showed that the infinity of both was equal. He showed that the infinite number of points in a line as long as your finger was equal to the infinite number in a line as long as from here to the Sun. In fact, the number of points on a line is equal to the number of points in a plane. And yet, some infinities are bigger than others. This is strange territory indeed, and requires some concentration to understand and enjoy, even with the Kaplan's literate, witty, and clear explanations. This is a fine introduction to different aspects of serious mathematics; true to its subtitle, it is a book full of pleasures. ... Read more

Book Description

Unusually clear and interesting classic covers real numbers and sequences, foundations of the theory of infinite series and development of the theory (series of valuable terms, Euler’s summation formula, asymptotic expansions, other topics). Exercises throughout. Ideal for self-study.
... Read more

Reviews (4)

How to tell if you are a "formalist" --
Anything to do with "infinity" is fascinating. Much of the history of mathematics has been a duel between those who see "infinity" as a delusion and impediment to progress, and those who see it as the greatest tool in the mathematician's toolbox. Infinite series, which may be loosely defined as sums of an infinite number of terms (numbers), take on some of this fascination. Although this book will appeal mainly to the professional mathematician, there is enough historical and elementary material to profit many college students- and possibly even some high school students.

Professional mathematician will find this book useful for filling in gaps left by topics not covered in traditional courses. An example is the detailed discussion of Euler's summation formula, which goes far beyond the simplified form usually encountered in textbooks. Another fascinating topic covered is divergent series, and methods by which meaningful sums can be assigned to these. There is something counterintuitive -- and, frankly, mind-boggling -- about many of these results.

Mathematicians can be put into several categories: 1) applied-mathematicians/computer-scientists/engineers concerned with solving practical problems, 2) those concerned with pedagogy and the history of mathematics, 3) epistemology and rigorous proofs, and 4) formalists. The fourth category, formalists, is difficult to define, but may be described as those that emphasize obtaining new results through formal (technical) manipulations, without undue concern regarding the meaning of the intermediate steps. The greatest exponents of this art were Euler and Ramanujan, though Fourier, Dirac and Heaviside are also solid members of this camp.

I take this digression because I feel that this book mainly appeals to the fourth type of mathematician. Although there are some general results in the theory of infinite series, any competent mathematicians can, in a few minutes, write a dozen infinite series which defy summation. As an example, the series associated with the Riemann zeta function of EVEN arguments were first summed by Euler. The sums arising from ODD arguments have defied summation to this day. Why this should be so is intriguing, but unknown. Incidentally, Euler's method of summation will make a "rigorists" hair stand upon ends. But he got the job done!

This is an ok book
To be honest alot of the work does not make immediate sense. Knopp leaves alot of important details out in his proofs and sometimes tends to rite to informal a proof of theorems. Theres no answers to the questions and i found that i became bored while reading the book. There are better. I have a large collection of books and the infinite series sections or chapters in them are better than alot of this book. I think that its main problem is that it cannot be easily accessible to beginners although it claims this. well apart from the begin chapters.

A Truly Marvelous Value
The last chapter on the Euler-MacLauren summation formula, and attendant interrelations among the Zeta function, Bernoulli Numbers and Bernoulli Polynomials is alone worth three times the price of this gem. Chock full of recipes and explanations of many of those little annoying points you don't understand fully. Do you REALLY understand what 'asymptotically equal to (~)' means? Heartily recommended!

The best book on infinite series
Excellent book for consulting with lots of examples and problems. Very well written but with the problem of very old notation. Everything you need to know about series is in this book. Very good to use in problems seminars ... Read more

Book Description

Combining concepts of mathematics and computer science, this book is about the sequences of symbols that can be generated by simple models of computation called "finite automata".Suitable for graduate students or advanced undergraduates, it starts from elementary principles and develops the basic theory. The study then progresses to show how these ideas can be applied to solve problems in number theory and physics. ... Read more

Reviews (1)

It will be the prime resource of the area for years to come
In 1983 Lothaire's "Combinatorics on Words" became the definitive resource on the area of stringology. 20 years later, Jean-Paul Allouche and Jeffrey Shallit's "Automatic Sequences" is set to be its heir. Pulling in countless concepts from many seemingly dissimilar disciplines, Allouche and Shallit are successful in bringing them together in an extremely informative and concise way. With detailed chapter notes and an immense bibliography the possible areas of exploration on each topic prove endless. A great resource for everyone from the serious researcher to the casually interested, this text is applicable to almost every area of mathematics and computer science. As the authors put it, "Sequences, both finite and infinite are ubiquitous in mathematics and theoretical computer science." If quality were synonymous popularity then this book will be equally ubiquitous. ... Read more

Book Description

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, PDE's, probability, diffraction, musical tones, and wavelets. Providing unified development of (univariate) Fourier analysis for functions on R, T, Z, and P, the book also includes an unusually complete presentation of the Fourier transform calculus. It uses concepts from calculus to present an elementary theory of generalized functions. It also uses the FT calculus and generalized functions to study the (univariate) wave equation, diffusion equation, and diffraction equation. In addition, fine points of the theory are developed. The book also demonstrates real-world applications of Fourier analysis in the chapter on musical tones.A valuable reference on Fourier analysis for a variety of scientific professionals, including Mathematicians, Physicists, Chemists, Geologists, Electrical Engineers, Mechanical Engineers, and others. ... Read more

Reviews (2)

Elegant, Sophisticated Math, Great Examples & Problems
I am taking this class as an undergraduate course with Dr. Healy at University of Maryland at College Park. This is an elegant, thoughtful book that provides a rich math course that is a welcome alternative to run-of-the mill engineering math classes lacking in intangible qualities. Many of the problems are tough and require some rigorous math at an advanced, math-major level (real analysis would be helpful) but are overall accessible to engineering seniors with strong math skills and office hours support from the professor, and the problems are well-geared to illustrating and exploring the topics in the text. As another benefit to the student, the examples definitely help one warm up for the problem solving at the end of the chapters. In addition to the richness and elegance of the subject as presented, this is a thoughtfully constructed and presented text.

The first several chapters introduce fourier transforms and related math such as convolutions as a set of operations in a variety of spaces, including continuous, discrete and periodic spaces. Then the text goes into the theory of distributions/generalized functions and solutions of differential equations. Several additional chapters take the subject into wavelets. The presentation of the Fourier transforms having a variety of manifestations in different kinds of spaces unifies in a fundamentally harmonious (no pun intended!) and beautiful way the disjoint and arbitrary Fourier processing taught to engineering undergraduates.

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 taking a modern computer approach to this ever relevant subject. Important topics such as sampling theory and the Fast Fourier Transform (FFT) are well covered and explained in detail. Also, chapters that apply Fourier Analysis to important physical areas (heat conduction, light diffraction, wave propagation, musical sound, etc.) illustrate and higlight the relevance of Fourier Methods in the real worls. There is also a nice summary at the end of the book that explains the histoy and most important application of Fourier Analyis (very nice). Ample computer excerices and the traditional proof/derivation homework problems are included. The book also seems to prepare the reader well for the increasingly subject of Wavelets and applying them musical sound. Also, what makes the book stand out from more traditional ones is the emphasis on Numerical Method and using the computer to solve or illustrate some of the powers of Fourier Analysis. Readers considering using this text should best have a background in calcus, differential equations and Matrix methods. This probably puts it at the junior/senior undergradudate level. 1st year graduate students might also benefit from the text.

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

Book Description

With much material not previously found in book form, this book fills a gap by discussing the equivalence of signal functions with their sets of values taken at discreet points comprehensively and on a firm mathematical ground. Mathematicians, physicists, and communications engineers will welcome the scope of information found here. ... Read more

Book Description

Wavelets: A Tutorial in Theory and Applications is the second volume in the new series WAVELET ANALYSIS AND ITS APPLICATIONS. As a companion to the first volume in this series, this volume covers several of the most important areas in wavelets, ranging from the development of the basic theory such as construction and analysis of wavelet bases to an introduction of some of the key applications, including Mallat's local wavelet maxima technique in second generation image coding. A fairly extensive bibliography is also included in this volume.

Key Features
* Covers several of the most important areas in wavelets, ranging from the development of the basic theory, such as:
* Construction and analysis of wavelet bases, and
* Introduction of some of the key applications, including Mallat's local wavelet maxima technique in second generation image coding
* Extensive bibliography is also included in this volume
* Companion to the first volume in this series, An Introduction to Wavelets, and can be used as supplementary instructional material for a two-semester course on wavelet analysis ... Read more

Reviews (1)

Slightly outdated but solid book - more theory than apps
As the title indicates, this book has some good, solid papers on wavelets/wavelet analysis. The title is a bit misleading -- the book is very theoretical and few of the papers address how wavelet techniques can be used to solve scientific and engineering problems. This is not to say that the book is not valuable -- it is edited by Chui, an extremely successful organizer of conferences, who has extensive experience writing and editing very good books. It is a mathematician's book.

At the time of its publication, this book was one of the better applications books, the other two comparable applications books were edited by: (1) Combes et al. (2) Ruskai et al.. This book has many more references to relevant works than the Combes and Ruskai books.

Unfortunately, 8 years have passed since the publication of this book, and a lot of progress takes place in the field of wavelets. Some techniques which looked promising 8 years ago did not live up to expectations, and some new ideas with very practical applications are not addressed in this book. This second remark may not be entirely fair since applications of wavelets have become so widespread that it would be impossible for any one book to cover all.

This book does not contain problems to assign for homework. It might be appropriate to use some chapters in the book for supplementary reading. in a graduate course or in an upper division special topics course. Unfortunately, the price is a bit high to make it required reading as a supplementary text for a course.

Some of the the theoretical papers, particularly the first 3 in Section I would not be of interest to applications oriented people since no work has followed from these papers. They are very specialized theoretical works.The papers in Section II cover topics which have become a standard tools in wavelet analysis, e.g., splines, biorthogonal wavelets, multiresolution analysis (MRA).Papers in Section III are a bit outdated. There was a period when wavelets were hyped and people in the research community believed that they could lead to improvements in the solution of PDEs. Unfortunately, few new practical tools have emerged. Only the paper on MRA in section IV may be of interest to a wide audience, but it too is rather theoretical. Sections V VI and VII were intended to cover applications to digital signal processing (DSP). If the primary interest of a reader is DSP, this is the wrong book to purchase in this day and age. Mallat's book: a Wavelet Tour of Signal Processing is far more comprehensive and up-to-date. ... Read more

Book Description

This book presents the theory and applications of Fourier series and integrals, Laplace Transforms, eigenfunction expansions, and related topics.It deals almost exclusively with those aspects of Fourier analysis that are useful in physics and engineering.Using ideas from modern analysis, it discusses the concepts and reasoning behind the techniques without getting bogged down in the technicalities of rigorous proofs.A wide variety of applications are included, in addition to discussions of integral equations and signal analysis. ... Read more

Reviews (3)

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.

The best book on the subject
Everything written in the above review is true and more:
The book gives the reader the connections between functional analysis, fourier analysis and PDE .the book is very intuitive and clear, and gives the reader the right picture on the subject. for the content of the book see the "table of contents" icon.
Read it and enjoy it!!.

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