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Wavelets Through a Looking Glass: The World of the Spectrum (Applied and Numerical Harmonic Analysis), by Ola Bratteli, Palle Jorgensen
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? Concise background material for each chapter, open problems, exercises, bibliography, and comprehensive index make this work a fine pedagogical and reference resource.; New previously unpublished results appear on the homotopy of multiresolutions, approximation theory, the spectrum and structure of the fixed points of the associated transfer, subdivision operators; Key topics of wavelet theory are examined; Excellent graphics show how wavelets depend on the spectra of the transfer operators; The important role of the spectrum of a transfer operator is studied; This self-contained book deals with important applications to signal processing, communications engineering, computer graphics algorithms, qubit algorithms and chaos theory.
- Sales Rank: #2856432 in Books
- Brand: Brand: Birkhäuser
- Published on: 2002-07-12
- Original language: English
- Number of items: 1
- Dimensions: 10.00" h x .94" w x 7.01" l, 1.62 pounds
- Binding: Hardcover
- 398 pages
- Used Book in Good Condition
Review
"Mere words cannot adequately describe all the great features of the book...which has something for everyone of all mathematical persuasions.... This book has quite a different perspective from the other monographs on wavelets...mainly because it emphasizes the Fourier domain as the proper "window" or "looking glass" from which one can most easily study wavelet theory.... Whatever his or her level of expertise with the subject, a reader of this book will never be bored.... Each chapter begins with an introductory section accessible to the layperson, often containing some historical background, and often extremely entertaining.... This is a fun book, full of exciting new results, written by two world-renowned experts in the field, which makes connections between a variety of important areas in pure and applied mathematics."
―SIAM Review
"This is a quite unusual wavelet book, with a fresh view on the subject. While the vast majority of wavelet books concentrate on multiresolution analysis and its applications, this book treats the topic from an operator theoretic point of view, with a focus on techniques having a geometric or spectral theoretic flavor. In fact, while parts of the book could be used in a wavelet course, other parts would be suitable in a course directed towards operator theory."
―Zentralblatt Math
"The authors take a fresh look at the subject and develop a new intuition for many topics. In particular, they make more extensive use of spectral theory than is usual in the subject. Each [chapter has] an introduction to the topics in the chapter and explains why they are discussed and where they come from. There are connections with many other areas of mathematics and physics not usually associated with wavelet theory . . . The end of each chapter has a large selection of problems in which much of the standard theory can be worked out by the reader. There is also a glossary of useful terms at the end of Chapter 1 where many of the terms . . . are clearly defined both from a mathematical and an applications point of view."
―Mathematical Reviews
"This book does a superlative job of demonstrating the richness of the theory of wavelets . . . The authors have demonstrated further connections with spectral theory, ergodic theory, homotopy theory and the theory of probability . . . At the same time the material is beautifully documented by means of . . . figures . . . and illustrations . . . The pedagogy is further enhanced by several paragraphs of illuminating prose at the beginning of each chapter . . . Although not written as a conventional text, . . . an industrious graduate student could profit enormously from exposure to this book. With respect to the literature on wavelets, it is difficult to recall any other book that is so well documented both with graphical and numerical details as well as mathematical proof. This volume will remain a central work for many years to come."
―Mark Pinsky, Northwestern University
"This book by Bratteli and Jorgensen serves as an introduction to the theory of wavelets and also a bringing up to date with the latest research problems. While there are many good books on wavelets, 'Wavelets Through a Looking Glass' brings a fresh perspective on the field. The theory is covered from the point of view of operator theory and functional analysis, with an emphasis on the connection between the spectral properties of the operators on the discrete data and the geometric features of the wavelet functions in the continuous counterpart. The book includes the now classical results of wavelet theory, such as the multiresolution construction and the correlation to the transfer operator, as well as some new results of the authors (published here for the first time), such as the index theorem or several spectral preperties of the transfer operator." --- J. Operator Theory
"The book gives a general persetnation of some recent developments in wavelet theory, with an emphasis on techniques that are both fundamental and relatively timeless haveing a geometric and spectral-theoretic flavour. The exposition is clearly motivated and unfolds systematically aided by numerous graphics. Excellent graphics show how wavelets depend on the spectra of the transfer operators. Some new results are presented for the first time." ---EMS Newsletter
Most helpful customer reviews
12 of 12 people found the following review helpful.
Best new book on wavelets!
By Mark A. Pinsky
Reader's review of "Wavelets Through A Looking Glass--The World of the
Spectrum", by O. Bratteli and P. Jorgensen
This book does a superlative job of demonstrating the richness of the
theory of wavelets, which began as an outgrowth of classical harmonic
analysis. The authors have demonstrated further connections with spectral
theory, ergodic theory, homotopy theory and the theory of
probability---just to name a few of the well-established areas of
mathematics which are shown to touch the theory of wavelets. At the same
time the material is beautifully documented by means of 61 figures,
numerous tables and other illustrations which are freely distributed
throughout the book. An exhaustive list of 200 references to the most
current literature ensures scholarly care and the most up-to-date account
of the topics covered.
The authors succeed admirably in achieving the two-fold purpose of the
book. On the one hand the goal is to give a modern (but "timeless")
presentation of wavelet theory while on the other hand the goal is to
present new results that have not previously been published. The latter
include material on homotopy of resolutions, approximation theory and
results on the spectrum of the associated transfer operators and
subdivision operators. The first goal is well-served by the many
well-documented exercises which appear at the end of each chapter. The
pedagogy is further enhanced by several paragraphs of illuminating prose
at the beginning of each chapter--to set the stage for the technical
material to follow. Although not written as a conventional text, one would
expect that an industrious graduate student could profit enormously from a
serious exposure to this book.
With respect to the literature on wavelets, it is difficult to recall any
other book that is so well documented both with graphical and numerical
details as well as mathematical proof. This volume will remain a central
reference work for many years to come.
Mark Pinsky,
Northwestern University
...
13 of 14 people found the following review helpful.
A fresh look at wavelets
By Akram Aldroubi
The Book by Bratteli and Jorgensen is a superb book on wavelet's theory. It is very well written and has new and a fresh point of view on the subject. Although there are several good books on wavelets, the book by Bratteli and Jorgensen covers an important niche that has not been covered before. In particular
1- The book covers the theory of wavelets from the point of view of operators and functional analysis and will appeal to a growing number of pure as well as applied mathematicians interested in the subject.
2- The writing of the book is very appealing: every chapter starts by a tutorial that gives motivation as well as intuition. It is then followed by a very clean mathematical development of the subject, together with many examples, figures, and applications from physics and engineering. A set of nice problems is provided at the end of each chapter. Thus this book can be used as a graduate textbook or for mathematical seminars in mathematics departments.
3- This book can even be used by experts in wavelet theory for learning about recent developments and new perspectives from operator theory and functional analysis.
I highly recommend this book.
13 of 14 people found the following review helpful.
A masterpiece in wavelets
By Ivan Avramidi
This is a very well written book on the theory of wavelets and its applications. The presentation is self-contained and may serve as an introduction for someone who wants to learn about this topic. It also is an excellent book for those who have an advanced graduate degree in applied mathematics since it demonstrates how to truly understand complex concepts. This book gives a general presentation of some recent developments in wavelet theory with an emphasis on techniques that have a geometric and spectral-theoretic flavor. It can be certainly used as a textbook for graduate students as well as as a reference book for the specialists and researchers. A very nice feature of the book are the tutorials at the beginning of each chapter and some sections, which serve as summaries of main ideas and guides through the background ideas and motivation. At the end of every chapter there are plenty of excercises of various level of difficulty. Overall, this is a great book and I recommend it highly to a motivated reader.
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