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Handbook of Fourier Analysis & Its Applications ReviewI don't write many reviews, but this is an unusual book.If you are looking for a conventional treatment of Fourier analysis, this is not it. The treatment is not only clear, but refreshingly creative. Marks is talented at explaining complex ideas in simple terms. He uses figures. LOTS of figures. Despite the clarity of presentation, there is no loss of foundational rigor. It is all there.
Marks is a very bright polymath and this book shows an incredible block of material. It would seem that the book has been developed over a number of years to reflect the particular lectures he has given as he taught graduate level courses on signal analysis, multivariate Fourier analysis, image processing, Shannon sampling theory, image compression, associative memories, etc. What I find to be most valuable is the relatively "new" material (from the past two decades) on multivariate Fourier analysis and its applications and on sampling, signal recovery, image processing ... areas where Marks has a deep knowledge of the literature. Likewise, the chapter on alternating projection onto convex sets bubbles over with fascinating applications like subpixel resolution, brachytherapy treatment of prostate cancer, recovery of lost portions of digital images, tomography, and image deblurring.
A LOT of new ideas are presented in unique ways. I particularly like the author's problem sets; he includes many of the solutions in the book. Many problems are not rote exercises from the chapters, but introduce new ideas from incisive perspectives. Who knew all elementary finance could be modeled as a convolution?
The author writes clearly and the illustrations are numerous, nicely done, and very helpful. I found that I could "learn" new ideas by studying sections of the book that were unfamiliar to me. And I found that the author displayed a mature up to date knowledge of the topics with which I am thoroughly acquainted. It reminded me of reading the encyclopedia when I a kid. I would start looking up one thing, something else would catch my interest, and I would read about it instead.
This book claims it can be used in junior and senior level courses such as Signals and Systems, Digital Signal Processing, and graduate level courses related to Fourier analysis such as Adaptive Signal Processing (on the topic of power spectral analysis), Probability Theory and Random Process (on the topics of characteristic functions of random variables), and Multidimensional Signal Processing.
There are imperfections. I've found a half dozen typos which doesn't seem bad for a book this thick. The bibliography looks a bit bloated and an Appendix listing of obscure probability models seems out of place. There were a few topics, like multidimensional IIR filters, that are fundamental but not treated. I guess you can't have everything.
Overall, this is a great book by a great author. Currently the material covered is scattered in different books or papers. It is very nice to see such an accessible volume put together by an expert in the field.Handbook of Fourier Analysis & Its Applications OverviewFourier analysis has many scientific applications - in physics, number theory, combinatorics, signal processing, probability theory, statistics, option pricing, cryptography, acoustics, oceanography, optics and diffraction, geometry, and other areas.In signal processing and related fields, Fourier analysis is typically thought of as decomposing a signal into its component frequencies and their amplitudes.This practical, applications-based professional handbook comprehensively covers the theory and applications of Fourier Analysis, spanning topics from engineering mathematics, signal processing and related multidimensional transform theory, and quantum physics to elementary deterministic finance and even the foundations of western music theory.As a definitive text on Fourier Analysis, Handbook of Fourier Analysis and Its Applications is meant to replace several less comprehensive volumes on the subject, such as Processing of Multifimensional Signals by Alexandre Smirnov, Modern Sampling Theory by John J. Benedetto and Paulo J.S.G. Ferreira, Vector Space Projections by Henry Stark and Yongyi Yang and Fourier Analysis and Imaging by Ronald N. Bracewell. In addition to being primarily used as a professional handbook, it includes sample problems and their solutions at the end of each section and thus serves as a textbook for advanced undergraduate students and beginning graduate students in courses such as:Multidimensional Signals and Systems, Signal Analysis, Introduction to Shannon Sampling and Interpolation Theory, Random Variables and Stochastic Processes, and Signals and Linear Systems.
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