Book Description

John Wallis was appointed Savilian Professor of Geometry at Oxford University in 1649. He was then a relative newcomer to mathematics, and largely self-taught, but in his first few years at Oxford he produced his two most significant works: De sectionibus conicis and Arithmetica infinitorum. In both books, Wallis drew on ideas originally developed in France, Italy, and the Netherlands: analytic geometry and the method of indivisibles. He handled them in his own way, and the resulting method of quadrature, based on the summation of indivisible or infinitesimal quantities, was a crucial step towards the development of a fully fledged integral calculus some ten years later.

To the modern reader, the Arithmetica Infinitorum reveals much that is of historical and mathematical interest, not least the mid seventeenth-century tension between classical geometry on the one hand, and arithmetic and algebra on the other. Newton was to take up Wallis’s work and transform it into mathematics that has become part of the mainstream, but in Wallis’s text we see what we think of as modern mathematics still struggling to emerge. It is this sense of watching new and significant ideas force their way slowly and sometimes painfully into existence that makes the Arithmetica Infinitorum such a relevant text even now for students and historians of mathematics alike.

Dr J.A. Stedall is a Junior Research Fellow at Queen's University. She has written a number of papers exploring the history of algebra, particularly the algebra of the sixteenth and seventeenth centuries. Her two previous books, A Discourse Concerning Algebra: English Algebra to 1685 (2002) and The Greate Invention of Algebra: Thomas Harriot’s Treatise on Equations (2003), were both published by Oxford University Press.

Reviews (3)

Not worth it. Definitely overpriced.
Too simplistic. Perhaps good for newbies. But anyway there
are better books on FFT, e.g. Brigham. This one is not for
professionals! You've been warned. For beginners I would recommend books by Zonst. They are much more to the point.
And cheaper! Thanks god I didn't buy it myself!

Simple yet concise
This book is a great way to realize that theoretical murmurings can mature into intuitive processes that can be used as guides to self-check new ideas. While FFT theory is necessary, if you like that kind of thing, a good grasp of what may be accomplished is often left out or buried. This book provides a great way to get perspective on rigorous frequency domain textbook theory and make the necessary textbook search for detail much more meaningful. I highly recommend this book for anyone who has to deal with signal processing.

Ramirez balances theoretical & practical aspects of FFTs.
Of the half-shelf of books in my library related to signal processing, this one by Ramirez is part of a highly valued subset. The author achieves an artful balance of the theoretical with the practical. Given that my emphasis is on applying the principles of FFT processing in a manufacturing environment, his approach proved quite helpful.

Throughout the book, basic concepts are initially presented within a mathematical context. Subsequently, they are reinforced with examples. Often, hardcopies of traces from a signal processor's monitor are provided--these are effective in driving home key points.

After reading this book over two days' time, I was left wishing to see a companion volume that adds more on real-life problem solving, plus material on advanced techniques (e.g., cepstrum). This thirst for more knowledge says a lot for the way in which Ramirez motivates. ... Read more

Book Description

Over the last two years, wavelet methods have shown themselves to be of considerable use to harmonic analysts and, in particular, advances have been made concerning their applications. The strength of wavelet methods lies in their ability to describe local phenomena more accurately than a traditional expansion in sines and cosines can. Thus, wavelets are ideal in many fields where an approach to transient behaviour is needed, for example, in considering acoustic or seismic signals, or in image processing. Yves Meyer stands the theory of wavelets firmly upon solid ground by basing his book on the fundamental work of Calderón, Zygmund and their collaborators. For anyone who would like an introduction to wavelets, this book will prove to be a necessary purchase. ... Read more

Book Description

This book provides a thorough and self-contained study of interdependence and complexity in settings of functional analysis, harmonic analysis and stochastic analysis. It focuses on "dimension" as a basic counter of degrees of freedom, leading to precise relations between combinatorial measurements and various indices originating from the classical inequalities of Khintchin, Littlewood and Grothendieck. Topics include the (two-dimensional) Grothendieck inequality and its extensions to higher dimensions, stochastic models of Brownian motion, degrees of randomness and Fréchet measures in stochastic analysis. This book is primarily aimed at graduate students specializing in harmonic analysis, functional analysis or probability theory. It contains many exercises and is suitable as a textbook. It is also of interest to computer scientists, physicists, statisticians, biologists and economists. ... Read more

Book Description

Harmonic analysis plays an essential role in understanding a host of engineering, mathematical, and scientific ideas. In Harmonic Analysis and Applications, the analysis and synthesis of functions in terms of harmonics is presented in such a way as to demonstrate the vitality, power, elegance, usefulness, and the intricacy and simplicity of the subject. This book is about classical harmonic analysis - a textbook suitable for students, and an essay and general reference suitable for mathematicians, physicists, and others who use harmonic analysis.Throughout the book, material is provided for an upper level undergraduate course in harmonic analysis and some of its applications. In addition, the advanced material in Harmonic Analysis and Applications is well-suited for graduate courses. The course is outlined in Prologue I. This course material is excellent, not only for students, but also for scientists, mathematicians, and engineers as a general reference. Chapter 1 covers the Fourier analysis of integrable and square integrable (finite energy) functions on R. Chapter 2 of the text covers distribution theory, emphasizing the theory's useful vantage point for dealing with problems and general concepts from engineering, physics, and mathematics. Chapter 3 deals with Fourier series, including the Fourier analysis of finite and infinite sequences, as well as functions defined on finite intervals. The mathematical presentation, insightful perspectives, and numerous well-chosen examples and exercises in Harmonic Analysis and Applications make this book well worth having in your collection. ... Read more

Reviews (3)

Bridging math and engineering.
I used this book in my teaching of a beginning grad level math (harmonic) analysis course, but relying on three books in all. Harmonic analysis is a big subject with many points of view (infinite in all directions, if you will!). One of the directions is the link to engineering problems. Specific areas of math often serve as service courses for engineers; and harmonic analysis is one area with a rich set of links to engineering problems. Things are complicated however by the the difference in terminology which is used in the two worlds. You might almost say that they speak different languages;-and they have different aims, by necessity! --Benedetto's book makes a heroic, and, in my view, very sucessful effort in highlighting the engineering significance of the basic principles of harmonic analysis. Few harmonic analysis books even try. Benedetto's book may be a bit hard for beginning students(there is a lot in it!), but the investment my students put into it was well worth the effort.

A Mysterious Book
I am not really sure if this book is good or not; for an electrical engineering student this book might as well be written in another language. Being a senior I had already covered about 50% of the material in this book in previous classes; I found it amazing that despite this, I could not confirm my knowledge by reading the same material out of the Benedetto book. The probalem is that the text is written EXTREMELY cryptically. I am not sure if this is standard notation for math students; but the notation used is incredible unfamilar and irratating for any of the engineering fields. In addition the book appears to lack any type of contunity. Although the chapters are laid out nicely, the inter-chapter organization seems to be comprised of a series of random theorms thrown together. Conclusion: In reading this book I had to "stop and translate" every few lines, the notation is not standard for engineering, and this book feels more like a series of scholarly papers tied together, than like a real book.

The book is not useful unless you're already a mathematician
After struggling with this book for a semester, it became apparent to me that while the book is rich in scope, its allusions to so many branches of mathematics and incomplete detailing of these allusions merely serve to bewilder the undergraduate and rookie grad. student of mathematics. Indeed, the book is an excellent reference for those who are already EXTREMELY well-versed in the subject and/or are looking to branch out into more detail and abstraction. For the rest of us, however, this book is ENTIRELY unsuitable--even, I daresay, for the "advanced undergraduate" this book purports to aim itself at. So ignore the hype of professional book reviewers and take it from someone in the trenches: buy something intelligible from the myriad of books out there on Fourier Theory and Harmonic Analysis. ... Read more

Book Description

This volume consists of two parts. Chapters 1-8, Basic Wavelet Analysis, are aimed at graduate students or advanced undergraduates in science, engineering, and mathematics. They are designed for an introductory one-semester course on wavelets and time frequency analysis, and can also be used for self-study or reference by practicing researchers in signal analysis and related areas. The reader is not presumed to have a sophisticated mathematical background; therefore, much of the needed analytical machinery is developed from the beginning. The only prerequisite is a knowledge of matrix theory, Fourier series, and Fourier integral transforms. Notation is introduced which facilitates the formulation of signal analysis in a modern and general mathematical language, and the illustrations should further ease comprehension. Each chapter ends with a set of straightforward exercises designed to drive home the concepts.

Chapters 9-11, Physical Wavelets, are at a more advanced level and represent original research. They can be used as a text for a second-semester course or, when combined with Chapters 1 and 3, as a reference for a research seminar. Whereas the wavelets of Part I can be any functions of "time," physical wavelets are functions of space-time constrained by differential equations. In Chapter 9, wavelets specifically dedicated to Maxwell's equations are constructed. These wavelets are electromagnetic pulses parameterized by their point and time of emission or absorption, their duration, and the velocity of the emitter or absorber. The duration also acts as a scale parameter. We show that every electromagnetic wave can be composed from such wavelets. This fact is used in Chapter 10 to give a new formulation of electromagnetic imaging, such as radar, accompanied by a geometrical model for scattering based on conformal transformations. In Chapter 11, a similar set of wavelets is developed for acoustics. A relation is established at the fundamental level of differential equations between physical waves and time signals. This gives a one-to-one correspondence between physical wavelets and a particular family of time wavelets. ... Read more

Reviews (8)

Very thorough treatment, but *not* a beginner's text.
A classic text *but* look at the graph on the cover very carefully because it is the only one you'll get until you reach the last chapters of the book. In terms of the presentation of the basic theorems and equations, the text is excellent ... BUT ... there is precious little to guide a student to an intuitive and practical understanding of the theory. In practice, one looks at the graphical representations of wavelets quite often -- just like the ones used for Fourier analysis (can you imagine your first Fourier analysis class with no graphs of convolution, impulses, etc? Wavelets *are* Fourier analysis taken a step further in order to handle time varying systems. Graphs are essential to the uninitiated.) In fact, it is helpful to think of wavelets (i.e., wavelet packet decompositions) as a Fourier spectrum with an additional time axis added so you can see how the spectrum changes over time. The "wavelet" method simply optimizes the resolution used at different frequencies. True, it is a little more complicated than that, but it is easy to lose sight of the simple elegance of the topic when so many equations are flung at you without intuitive context. The topic could be presented MUCH better, so don't worry that it takes a long time to get through this material. Read other basic wavelet texts first and then *definitely* come back to this one. You will appreciate its otherwise *excellent* presentation much more and the principles will be easier to put into actual practice rather than mere academic conjecture.

A clear introduction to the notation and theory of wavelets
Some reviewers have commented that this is not such a "friendly"
guide if you are not friends with upper division or graduate
level mathematics. There is some truth to this. Unlike the
pragmatic approach taken in "Ripples in Mathematics" this is
a mathematical coverage of wavelets.

For me the value in this book is that it provides a clear
introduction to the notation and theory behind wavelets. This
book provides the tools you need to understand the wavelet
literature better. If you are a software engineer searching
for a quick guide to wavelet algorithms, this book may disappoint.

Reader friendly.
I really did find this lovely book reader friendly! The author has succeeded in communicating the exciting subject of wavelets, and their many applications, to students and more advanced readers alike. He realized that different communities, math, computer science, engineering, and physics,--
that they have variations in their emphasis, their terminology,
and their thinking. When authors speak to the various user groups, and do it well,-- like in this case, the result is a cucess. This friendly book is!

Good theory but lacking applications
It is not a friendly guide to Wavelets. However, chapters 1 one to six are excelent. Chapter 7, Multiresolution Analysis, the heart of the matter is hard to understand. So, you get the flavor of Wavelets but not learn to use it.

Outstanding! Written for broad audience of potential users!
I bought this book when I was particularly interested in continuous wavelet transforms. I found myself flying through the exposition. I felt confident about the knowledge I was acquiring and I was quickly able to apply it. Although I come at this book as a mathematician, I think that it is ideal for engineers and physical scientists who usually have far better grounding in signal processing and related issues in Fourier analysis than do mathematicians. I have recommended this book to students, friends and colleagues with high praise. ... Read more

Book Description

This is a follow-on to Andy's first book - Understanding the FFT.It presents the fundamental mathematical notions underlying the DFT and FFT at essentially the same level as the first book.It goes on to illustrate applications of the FFT it instrumentation, audio and image enhancment (2-dimensional FFT), developing necessary peripheral techniques along the way. ... Read more

Book Description

From the reviews: "A good introduction to a subject important for its capacity to circumvent theoretical and practical obstacles, and therefore particularly prized in the applications of mathematics. The book presents a balanced view of the methods and their usefulness: integrals on the real line and in the complex plane which arise in different contexts, and solutions of differential equations not expressible as integrals. Murray includes both historical remarks and references to sources or other more complete treatments. More useful as a guide for self-study than as a reference work, it is accessible to any upperclass mathematics undergraduate. Some exercises and a short bibliography included. Even with E.T. Copson's AsymptoticExpansions or N.G. de Bruijn's Asymptotic Methods inAnalysis (1958), any academic library would do well to have this excellent introduction." (S. Puckette, University ofthe South) #Choice Sept. 1984#1 ... Read more

Book Description

Learn to juggle numbers! This book is the first comprehensive account of the mathematical techniques and results used in the modelling of juggling patterns. This includes all known and many new results about juggling sequences and matrices, the mathematical skeletons of juggling patterns.

Many useful and entertaining tips and tricks spice up the mathematical menu presented in this book. There are detailed descriptions of jugglable and attractive juggling sequences, easy zero-gravity juggling, robot juggling, as well as fun juggling of words, anti-balls, and irrational numbers.

The book also includes novel, or at least not very well known connections with topics such as bell ringing, knot theory, and the many body problem. In fact, the chapter on mathematical bell ringing has been expanded into the most comprehensive survey in the literature of the mathematics used by bell ringers.

Accessible at all levels of mathematical sophistication, this is a book for mathematically wired jugglers, mathematical bell ringers, combinatorists, mathematics educators, and just about anybody interested in beautiful and unusual applications of mathematics. ... Read more

Book Description

Wavelet theory had its origin in quantum field theory, signal analysis, and function space theory. In these areas wavelet-like algorithms replace the classical Fourier-type expansion of a function. This unique new book is an excellent introduction to the basic properties of wavelets, from background math to powerful applications. The authors provide elementary methods for constructing wavelets, and illustrate several new classes of wavelets. The text begins with a description of local sine and cosine bases that have been shown to be very effective in applications. Very little mathematical background is needed to follow this material. A complete treatment of band-limited wavelets follows. These are characterized by some elementary equations, allowing the authors to introduce many new wavelets. Next, the idea of multiresolution analysis (MRA) is developed, and the authors include simplified presentations of previous studies, particularly for compactly supported wavelets.Some of the topics treated include:oSeveral bases generated by a single function via translations and dilationsoMultiresolution analysis, compactly supported wavelets, and spline waveletsoBand-limited waveletsoUnconditionality of wavelet basesoCharacterizations of many of the principal objects in the theory of wavelets, such as low-pass filters and scaling functionsThe authors also present the basic philosophy that all orthonormal wavelets are completely characterized by two simple equations, and that most properties and constructions of wavelets can be developed using these two equations. Material related to applications is provided, and constructions of splines wavelets are presented. Mathematicians, engineers, physicists, and anyone with a mathematical background will find this to be an important text for furthering their studies on wavelets. ... Read more

Reviews (1)

The Hernandez-Weiss wavelet book
The Hernandez-Weiss book is a great source in mathematics, where the reader can learn about
some fundamental and timeless tools of wavelet analysis. The presentation is clear and self contained. The book is great for an upper level analysis course, as well as for self study. Its emphasis is centered around the
multiresolution method, and it has both the basics in the subject, and
some exciting new results,-- those due
to the co-authors, as well as those of the analysts in the extended "Wash U-family" of researchers. It has a wealth of new material that
came after the 1992 classic of I. Daubechies ("Ten lectures on wavelets" SIAM ).The presentation in Hernandez-Weiss of, for example,
multiplicity theory and the dimension function is great for
the student who wishes to pick up the central ideas, or to
learn the details, step by step. The book contains both history and
motivation, and it points the reader towards a variety of exciting and timeless applications. ... Read more

Book Description

This concise, self-contained reference text addresses all of the major topics in Fourier series--emphasizing the concept of approximate identities and presenting applications, particularly in time series analysis. ... Read more

Book Description

This volume will serve several purposes: to provide an introduction for graduate students not previously acquainted with the material, to serve as a reference for mathematical physicists already working in the field, and to provide an introduction to various advanced topics which are difficult to understand in the literature. Not all the techniques and application are treated in the same depth. In general, we give a very thorough discussion of the mathematical techniquesand applications in quatum mechanics, but provide only an introduction to the problems arising in quantum field theory, classical mechanics, and partial differential equations. Finally, some of the material developed in this volume will not find applications until Volume III. For all these reasons, this volume contains a great variety of subject matter. To help the reader select which material is important for him, we have provided a "Reader's Guide" at the end of each chapter. ... Read more

Book Description

A conversation between Euclid and the ghost of Socrates. . . the paths of the moon and the sun charted by the stone-builders of ancient Europe. . .the Greek ideal of the golden mean by which they measured beauty. . . Combining historical fact with a retelling of ancient myths and legends, this lively and engaging book describes the historical, religious and geographical background that gave rise to mathematics in ancient Egypt, Babylon, China, Greece, India, and the Arab world. Each chapter contains a case study where mathematics is applied to the problems of the era, including the area of triangles and volume of the Egyptian pyramids; the Babylonian sexagesimal number system and our present measure of space and time which grew out of it; the use of the abacus and remainder theory in China; the invention of trigonometry by Arab mathematicians; and the solution of quadratic equations by completing the square developed in India.These insightful commentaries will give mathematicians and general historians a better understanding of why and how mathematics arose from the problems of everyday life, while the author's easy, accessible writing style will open fascinating chapters in the history of mathematics to a wide audience of general readers. ... Read more

Reviews (3)

Wavelets: from valley to crest
Choosing a book on Wavelets to use as material for an introductory course is far from simple: there are many books on the subject and often the mathematical starting level of the audience is not uniform. But the book by Blatter is in my opinion very good if one wants to start from scratch (well, not quite of course: some knowledge of Lebesgue integration and infinite dimensional Banach and Hilbert spaces of functions is needed) in a mathematical or technical environment.

Starting with 2 chapters on Fourier analysis, going from windowed Fourier transform to a first introduction of continuous wavelet transform (CWT), and giving connections to several well known facts from signal analysis (the Heisenberg uncertainty priciple and the Shannon sampling theorem), the theoretical framework of the CWT is given in chapter 3. Interesting are the results on the decay of the wavelet transform in terms of the smoothness of the signal and the number of vanishing moments for the wavelet.

This is followed by a chapter on the important subject of 'frames' and one on multiresolution analysis (the foundation for the hierarchy of orthonormal bases for L^2(R)). The scaling function (time domain) is actuallly constructed from its properties in the Fourier domain (i.e. the frequency domain).

The final chapter treats the basic idea behind and constructions of orthonormal wavelets with compact support (including the Daubechies wavelets), also touching upon binary interpolation and so called spline wavelets.

For applications to a host of disciplines it is possible to consult one of the many applied books: in my opinion it is necessary to have a firm grasp of the underlying theoretical framework before really understanding what one is doing. I have nothing against applications, but once one has to go back to the roots: back to the valley to be able to reach the crest of the wavelet without accidents.

A real primer
This booklet, the translation of the original German version "Wavelets - Eine Einfuehrung" (published by Vieweg-Verlag), is indeed a real primer.

It removes old layers of thinking about the structure of signals and replaces them by an elegant formalism rooted in the theory of Hilbert spaces and orthonormal bases.

A graduate background on Hilbert spaces (L_2 etc.), the Fourier transform and some integration theory (Lebesgue integration) is recommended.

There are very few (easily corrected) typos and the book can be used as a basis for a one quarter course of 4 hours each week during 7 weeks (or a semester of 2 hours), with a backing of handout problems on several aspects of the theory.

Good on theory-bad on application
This book is a good tool for those who want to explore the mathematical rigors of wavelet theory. The "primer" assumes that the user is a fairly well versed mathematician, but the derivations are complete. There is little to nothing said about the practical applications of wavelets. For applications, look to other books. ... Read more

Book Description

This book introduces 2-D wavelets via 1-D continuous wavelet transforms. The authors then describe the underlying mathematics before progressing to more advanced topics such as matrix geometry of wavelet analysis and three-dimensional wavelets. Practical applications and illustrative examples are employed extensively throughout, ensuring the book's value to engineers, physicists and mathematicians. Two-dimensional wavelets offer a number of advantages over discrete wavelet transforms, in particular, for analysis of real-time signals in such areas as medical imaging, fluid dynamics, shape recognition, image enhancement and target tracking. ... Read more

Book Description

This research monograph addresses recent developments of wavelet concepts in the context of large scale numerical simulation. It offers a systematic attempt to exploit the sophistication of wavelets as a numerical tool by adapting wavelet bases to the problem at hand. This includes both the construction of wavelets on fairly general domains and the adaptation of wavelet bases to the particular structure of function spaces governing certain variational problems. Those key features of wavelets that make them a powerful tool in numerical analysis and simulation are clearly pointed out. The particular constructions are guided by the ultimate goal to ensure the key features also for general domains and problem classes. All constructions are illustrated by figures and examples are given. ... Read more

Book Description

In this book, the authors present the newly developed theory of non-harmonic Fourier series and its applications to the control of distributed parameter systems, and they extend the theory to include vector exponential series. The first part of the book presents the modern theory of exponentials, using an operator theory approach. The second extends and upgrades the method of moments--one of the most powerful tools in the flourishing theory of the control of distributed parameter systems.They then go on to discuss the controllability of systems described by parabolic and hyperbolic PDE's for internal, boundary, initial, and pointwise control, and consider typical applications to optimal control problems. Researchers in control theory, operator theory, functional analysis, and partial differential equations will find much to interest them in this treatise. ... Read more

Book Description

This is an introductory volume on a novel theory of basic Fourier series, a new interesting research area in classical analysis and q-series. This research utilizes approximation theory, orthogonal polynomials, analytic functions, and numerical methods to study the branch of q-special functions dealing with basic analogs of Fourier series and its applications. This theory has interesting applications and connections to general orthogonal basic hypergeometric functions, a q-analog of zeta function, and, possibly, quantum groups and mathematical physics.Audience: Researchers and graduate students interested in recent developments in q-special functions and their applications. ... Read more