Wavelets Through a Looking Glass: The World of the Spectrum (Applied and Numerical Harmonic Analysis) 2002nd Edition by Ola Bratteli (PDF)

Description

? 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bibliography, and comprehensive index make this work a fine pedagogical and reference resource.; New previously unpublished results appear on the homotopy of multiresolutions, approximation theory, the spectrum and structure of the fixed points of the associated transfer,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.

User’s Reviews

Editorial Reviews: 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

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is easy to read, straight to the point, uses easy vocabulary and a pleasant style. As the book’s subtitle says “The world of the spectrum” it actually delivers what it says: the world of the spectrum for wavelets!. The looking glass the authors are referring to is Fourier transforms, as such the book take a beautiful well written easy to read journey into the wavelets world with all about spectrum analysis right from the very beginning with a looking glass through Fourier transforms. To me, the book has been a jewel and a rarity to hold in your hands; such a master piece. It helps a lot if you are familiar with spectrum analysis, eigenvalues, eigenvectors, and FFT and what Fourier transform is. But to my great delight the book also makes these topics seem approachable and quite in reach, the authors do not dive into unnecessary detail or diverge at all, and have managed to stay the course on what they want to offer and remain focused: wavelets, wavelets and more wavelets understanding in areas, views and perspectives that might not have been presented often times elsewhere. Bottom line, two very enthusiastic thumbs up for this book indeed.

⭐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.

⭐Reader’s review of “Wavelets Through A Looking Glass–The World of theSpectrum”, by O. Bratteli and P. JorgensenThis book does a superlative job of demonstrating the richness of thetheory of wavelets, which began as an outgrowth of classical harmonicanalysis. The authors have demonstrated further connections with spectraltheory, ergodic theory, homotopy theory and the theory ofprobability—just to name a few of the well-established areas ofmathematics which are shown to touch the theory of wavelets. At the sametime the material is beautifully documented by means of 61 figures,numerous tables and other illustrations which are freely distributedthroughout the book. An exhaustive list of 200 references to the mostcurrent literature ensures scholarly care and the most up-to-date accountof the topics covered.The authors succeed admirably in achieving the two-fold purpose of thebook. 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 topresent new results that have not previously been published. The latterinclude material on homotopy of resolutions, approximation theory andresults on the spectrum of the associated transfer operators andsubdivision operators. The first goal is well-served by the manywell-documented exercises which appear at the end of each chapter. Thepedagogy is further enhanced by several paragraphs of illuminating proseat the beginning of each chapter–to set the stage for the technicalmaterial to follow. Although not written as a conventional text, one wouldexpect that an industrious graduate student could profit enormously from aserious exposure to this book.With respect to the literature on wavelets, it is difficult to recall anyother book that is so well documented both with graphical and numericaldetails as well as mathematical proof. This volume will remain a centralreference 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 particular1- 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 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.

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