Customer Reviews

Wavelets through a Looking Glass (Applied and Numerical Harmonic Analysis)

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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
...

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.

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.

This is a book about an important topic in applied mathematics by two authors with excellent credentials in both pure and applied areas. The reader will find many intriguing threads connecting wavelets to other parts of mathematics, including a wavelet index theorem, quantum computing, the ubiquitous C*-algebras O_n and, of course, spectral theory. The graphics are meticulously done.

I look forward to learning a lot from it.

This is a book about an important topic in applied mathematics by two authors with excellent
credentials in both pure and applied areas.

The reader will find many intriguing threads connecting wavelets to other parts of mathematics, including a wavelet "index theorem", quantum computing, the ubiquitous Cuntz C*-algebras and, of course, spectral theory. The graphics are excellent. I look forward to learning a lot from it.

Spectral theory forms a natural environment for the study of wavelets. The use operator algebras is unique and interesting. The text is written such that the analysis is precise and clear throughout. The book reads nicely without being chatty or overly terse and tiresome. Many of tge proofs are extremely well laid out. The problems are carefully chosen (ranging from challenging to knock your socks off bruisers!) so get your erasers out. Mathematicians, computer scientists, theoretical physicists working in quantum computing will likely find this book useful. Recommended to anyone working with wavelets.
A little bit of functional analysis will come in handy for those of wanting to do a self study.

This is an important mathematical reference written in excellent style. Wavelets have found applications in many areas of engineering and CS. The authors provide a detailed, rich and entertaining tour through this relatively young but important field for both math and CS/Eng. Connections are, e.g., made between advanced CS virtual-reality applications such as audio-systems processing, future applications such as quantum computing, and advanced math in functional analysis and operator theory.