Fourier Analysis

The book is great, but was somewhat damaged when received.

Writing a book on Fourier theory is fraught with problems simply because of the technical difficulty of the subject and the enormous scope of applications. The 19th century roots of the theory were highly analytical and revolved around ever more subtle issues of convergence which persisted even until 1966, culminating with Lennart Carelson's famous proof. Applied mathematicians have used Fourier theory as a daily tool for many years without worrying too much about subtle convergence properties, Schwartz spaces and so on. Indeed, Fourier theory figured in the discovery of DNA. In fact Fourier theory is everywhere and Tom Korner's book gives a real flavour of how far its tentacles extend. His chapters on the transatlantic cable, the diameter of stars, the age of the earth, Brownian motion, scientific fraud and more of are considerable interest and it is not often that one finds so much relevant historical material thrown into a mathematical tract. But that is Tom Korner.For those who aren't aware, Korner is a Cambridge mathematician who has spent his life specialising in Fourier theory. He also has a sense of humour. He is fond of finding counter examples in analysis and those who have read his lecture notes (check out his website) will know exactly what I mean. He takes you to the edge and gently pushes you over with a smile. He brings a classical analysts' approach to Fourier analysis as a subject which also cross-dresses as a highly applied discipline. It is possible to do electrical engineering courses without knowing all the subtleties that give hard core Fourier analysts a stirring in their mathematical loins. Indeed, there is one course in a well known US university where the Sampling Theorem is proved using distribution theory without the students ever having to sully their hands with the squalid business of proving any of the underlying theory. The Rigour Police are on perpetual holiday. Not so with Korner - the Rigour Police pistol whip suspect theorems until they give up their secrets! Then there are the highly applied medical imaging courses which focus on practical applications of this wonderfully broad theory. The list goes on and on and you can't please everyone.It must be remembered that Korner has pitched this book at Tripos level candidates at Cambridge ie a very sophisticated audience. If you are a struggling B or C grade student at a lesser university you will probably find the book unappealing. He wants to self select those who are capable of doing a PhD in the subject at Cambridge or somewhere equivalent. He doesn't have to apologise for that. It just means his audience is small.Interestingly, unlike Eli Stein's development of undergraduate Fourier theory and functional analysis more generally at Princeton, and in his series of textbooks, Korner doesn't touch the Schwartz space with a barge pole. But Stein has different fish to fry and it suits him to go down that pathway. Once you are in Schwartz space you can do the mathematical equivalent of terrible things to small furry animals without getting arrested.Any way you cut it, this book is well worth reading and very informative and if you can fill in all the gaps that Korner teases you with, even better.

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.

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.

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.

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

Tom write both clear mathematics and tells wonderful stories about the theorems and mathematicians who discovered them. There is a lot more than than the standard Fourier applications in the book. As many chapters are independent of each other it is possible to dip into the middle of the book with ease.

Questo testo e' una sintesi perfetta fra l'analisi armonica più avanzata e l'analisi di Fourier " di base"; tutto e' corredato con simpatiche note relative ad alcuni matematici che hanno dato contributi al l'analisi di Fourier. Nel testo e' presente un'introduzione del Metodo Monte Carlo.L'unico "difetto" di questo libro e' solo il prezzo un po' alto.Assolutamente consigliabile

Toute la puissance et la richesse de l'analyse modale appliquée à de nombreux sujets très variés (théorie des nombres, analyse numérique, contrôle, statistiques, géologie, astronomie, ingénierie), exposées de manière très didactique et stimulante avec la rigueur et la précision des mathématiques, et une vision historique des développements tout à fait intéressante (notamment sur la pose du câble transatlantique).Pourquoi tous les livres scientifiques ne relatent-ils pas cette extraordinaire aventure intellectuelle de la compréhension du monde et des progrès qui en découlent ?

本書の訳書 フーリエ解析大全〈上〉 ， フーリエ解析大全〈下〉 を読み始めてすぐに，次次に現れる「誤植」と「誤訳を疑わせる記述」に，たまらず，本書を取り寄せました．ほぼ3分の1を訳書と突き合わせながら精読（最終部のDirichletの定理に重点を置いた）して気付いこと．1.訳書の〈訳者序〉には 「（訳を担当した）諸君には原著の数式のミスプリントなどの修正も御願いした」とあるが，原著の誤植（かなり多い！）の大半が見過ごされ，新たに大量の誤植が付け加えられている．2.訳出されていない脱落（パラグラフ丸ごと，文，語句，数式）が大量にある．3.数学の内容に関わる誤訳が大量にある．詳細は出版社に報告済み（指摘箇所は300項目余りに上る）です．とびきりの誤訳を紹介します．訳書下巻304頁11行目に「以上の議論を振り返って，iに点を打ち，tに横棒を引けば，次の定理が得られる．」とあります．本書（原著）では「dotting the i's and crossing the t's」（慣用句）です．そもそも本書には「点を打つ，横棒を引く」に当たる数学上の「操作」は登場しません．しかし，訳者は「`点を打ち横棒を引く'という〈おまじない〉で定理が得られる」と平然と言い放ちます．この例に見られるように，訳者達は多くの場面で「原著の内容を理解していない，理解しようとさえしていない」状態です．読者への配慮など微塵もあろうはずがありません．というわけで，訳者陣の〈学力不足とモラルの欠如の結晶〉とも言うべき訳書『フーリエ解析大全』で「数学を学ぶ」ことは不可能であり，本書（原著）で学びましょう，というのが結論です． フーリエ解析大全〈上〉フーリエ解析大全〈下〉