Customer Reviews

Fourier Analysis

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4 star:		(1)
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1 star:		(1)

 

 

This book makes great reading. There is a fair amount of (well written) high level mathematics, but also a number of sections of a more historical or narrative nature, and a wonderful sense of humor pervades the work. The account of the laying of the transatlantic cable in the nineteenth century and the technical problems associated with it is priceless. Several sections are devoted to the life of Fourier. There is also a companion volume entitled ``Exercises for Fourier analysis''.

I am a physician though I was a math grad student at MIT for 5 years a long time ago. I found this a wonderful book for reading and working though in spare time. I used to think of Fourier Series as engineering stuff so I was surprized to find Dedekind's theorem on primes in arithmetic progressions and Weyl's on the distribution of integral multiples of an irrational modulo 1. Korner succeeded in giving me an entirely new outlook on harmonic analysis as well as some new perspectives on surprizing, incident areas of math.

In addition, he is a fine writer. I would put him up with Halmos in quality though a very different style.

This book is valuable for its emphasis on interesting applications. The treatment of the mathematical basics of Fourier analysis is too hasty to be of much value as a first text, but this is only to leave room for the many beautiful applications. To set the tone, Weyl's equidistribution theorem appears on page 11 (sic). Later we see the classical problems of 19th century physics, but also little samples of Brownian motion, Monte Carlo methods, cryptography and other modern things. When Korner feels like it, he includes historical remarks and anecdotes that are pleasantly told. I wish this cosy atmosphere could also have been upheld in the mathematical details. Unfortunately, when it comes to the proofs, Korner seems to have a bit of a macho attitude towards long calculations. A few words of explanation here and there would probably be of great help when one is lost in page after page of technical lemmas consisting of nothing but formulas and curious integral approximations.

The author weaves together stories, application, and mathematics to give a fairly complete vision of Fourier analysis. All the mathematics is done without requiring measure theory, and the motivation is always in the forefront. My personal favorite parts are the on the building of the transatlantic cable and the example of outstanding statististical analysis. Make sure to purchase the exercises book, even if it is only for the jokes.

Wonderful book! Fourier was a man of immense and multivariate talent. His main mathematical discovery shares this property of him: besides its beauty and depth, its applications arise practically everywhere. The book of Korner is both rigorous and delightful. A very rare combination.

This book is definitely not for those who really want to learn Fourier analysis. As soon as you read the preface you start thinking it is going to be a bad book, as the author states it explcitily "this is not a book for this, not a book for that..." It sticks to a list of theorems without proof many times or with ugly proofs other times. It is not structured and not even explain the motivation behind each result.


If someone really wants to know Fourier anlisys I would recommend "Fourier and Wavelet Analysis" from Bachman, Narici and Beckstein