Fourier Analysis: An Introduction (Princeton Lectures in Analysis Book 1) by Elias M. Stein

Fourier Analysis: An Introduction (Princeton Lectures in Analysis Book 1) Reviews

• 
  William Bies

  Fourier analysis forms a staple topic for freshman engineers and physicists. For instance, electrical circuits with inductance and reactance lend themselves much more conveniently to understanding in the frequency domain. But the typical approach in courses for engineers and physicists concentrates on the mechanics of computing and manipulating Fourier series resp. transforms and does not go very deeply into the theory behind it. It may come as something of surprise, therefore, to learn that the theory itself turns out to be elegant and profound and, ever since its inception, always to have been a driving force behind the development of pure mathematics in its own right.
  
  The present work by Elias Stein and Rami Shakarchi entitled Fourier Analysis: An Introduction (Princeton University Press, 2003) represents the first in a four-volume series of lectures on analysis for advanced undergraduates. Stein was a professor and legendary teacher at Princeton University, Shakarchi his gifted protégé who left the field for the finance industry but agreed to stay on long enough to assist in converting Stein’s lecture notes into presentable textbooks. To judge by this first installment, the issue is happy, indeed – it forms an ideal introduction to the subject of Fourier analysis for the undergraduate mathematics major.
  
  The technical level should be approachable to almost anyone who will have completed an introductory course in analysis (what in the usual curriculum comes immediately after multivariate calculus). It relies on the Riemann integral throughout, for instance, and mentions Lebesgue integration only in passing. The idea a beginning student should conceive is to go through it rapidly in order to acquire the basics of Fourier analysis at a higher level than, say, Courant and John (cf. our review of the latter,
  here), but still within reason (postpone for the time being really advanced aspects such as the Fourier transform of tempered distributions in Reed and Simon). If one keeps the objective in mind, he may well derive much pleasure from Stein-Shakarchi’s supremely elegant treatment of the elements.
  
  For to justify Joseph Fourier’s contention that pretty much every periodic function ought to be expandable into a convergent Fourier series is easier said than done (in fact, Fourier himself never proves it in his pioneering monograph from 1822, the The Analytical Theory of Heat [Théorie analytique de la chaleur]). Before one could do so, mathematicians had to get clear on the underlying notions, such as what class of functions to entertain and in what sense to understand the convergence. At the time, the going concept of a function meant something that was piecewise analytic, but in fact one should want to extend it at least to continuous if not continuously differentiable functions, possibly having discontinuities on a set of measure zero (thus, Riemann-integrable). The German (despite his French-sounding name) mathematician Peter Gustav Lejeune Dirichlet was the first to develop a rigorous theory of Fourier series in 1829.
  
  Thus, after an introductory sketch of the subject in chapter one, Stein-Shakarchi aim to get this far in chapters two and three. The first topic they tackle is that of the uniqueness and convergence of Fourier series (that is, of their symmetric partial sums). Uniqueness is easily disposed of, but the question of convergence plunges one into a welter of complications. The partial sum is readily seen to be equivalent to convolution with the Dirichlet kernel; the mooted complications arise because the Dirichlet kernel itself fails to have the very desirable properties one would want of a good kernel. Thus, the investigation forks into two: first, one can seek out other kernels that would be good in the requisite technical sense, or second, try to develop the technique with which to deal with the Dirichlet kernel. The first leads to the concepts of Abel and Cesàro summability and culminates in Fejér’s celebrated theorem, which says that every Riemann-integrable periodic function has a Fourier series that is Cesàro summable at its points of continuity and, moreover, if continuous everywhere, the convergence will be uniform.
  
  Chapter two is a sheer pleasure to read and goes well beyond Courant and John even though they prove the same result in the end (convergence using Dirichlet kernel) but without the more general perspective (good families of kernels). The theory becomes clear and economical once one has all the right concepts at one’s disposal (Cesàro and Abel summability, convolution, kernels etc.).
  
  In chapter three, after functional-analytic preliminaries everyone will have seen already anyway, one proves the major theorem on mean-square convergence of Fourier series and the remarkable Riemann-Lebesgue lemma, which implies that our intuitive expectation that sufficiently high Fourier coefficients should tend to zero is satisfied in the case of Riemann-integrable functions, at least. Chapter three concludes with a worked example of a continuous function whose Fourier series diverges at a point.
  
  The concise chapter four delves into some application of Fourier series: Hurwitz’s surprisingly simple solution in 1901 of the isoperimetric inequality (that of all closed curves in the plane of a given length, a circle encloses the greatest area), Weyl’s equidistribution law and an explicit example of a continuous but nowhere differentiable function. The homework exercises to this chapter are harder; it can be delicate to extract the leading behavior of the sums resp. integrals in question, but of course therefore they make for good practice.
  
  Chapters five and six are devoted to the Fourier transform on the real line, resp., Euclidean space of any (finite) dimension. Here, it is clear one needs some condition on decay at infinity in order to ensure that the improper integrals become well defined. But just what to demand may not be obvious, at first. Stein-Shakarchi suggest what would be about the minimum, namely, a criterion of moderate decrease (falling off as 1/(1+x^(1+ε)) for any ε>0). Yet, as the French mathematician Laurent Schwartz realized in his trend-setting work during the mid-twentieth century, the theory goes through even better if one requires more, namely, rapid decrease in the sense now named after him. These conditions are sufficient to establish the existence of the Fourier transform and its nice properties with respect to convolution, multiplication, differentiation and so forth. The goals of chapter five are to prove the Fourier inversion formula, Plancherel’s theorem (i.e., unitarity), applications to the Weierstrass approximation theorem, the heat equation and Laplace’s equation (involving the heat resp. Poisson kernels) and the Poisson summation formula. Chapter six on the generalization to Euclidean space covers about what one would expect it to (and, as such, is not especially deep). It concludes, however, with a discussion of Huygens’ principle and the Radon transform. The latter, of course, is significant technologically for medical imaging devices. (As the authors show, in three dimensions, the inverse of the Radon transform is unique and effectively expressible in terms of the Fourier transform.)
  
  This reviewer decided to skip over the relatively brief chapters seven and eight on finite Fourier analysis with applications to number theory (the fast Fourier transform, abelian groups, zeta function and Euler product, Dirichlet characters and L-functions). Needless to say, finiteness of the sums renders the analysis unproblematical and the whole tenor of the discussion is oriented to other questions than those that figure in Fourier analysis on the circle or in Euclidean space. In view of how excellent Stein-Shakarchi’s exposition has been up to this point, however, no doubt these chapters would offer a fine introduction – maybe someday! (NB: the celebrated prime number theorem calls for knowledge of complex function theory and so goes beyond the scope of this volume, although Stein-Shakarchi will cover it in their second volume on complex analysis.)
  
  The homework problems at the end of each chapter are admirably good. Even the easy ones are ordinarily there to point out a phenomenon of intrinsic interest and are so only because Stein-Shakarchi strategically place them right after all the necessary tools have been developed, with a hint as to how to think about the problem in the right way. If it were not for this, one might well rack one’s brains for a long time without seeing how to solve them. A typical example would be this: it is impossible for a non-zero function and its Fourier transform both to be compactly supported.
  
  Stein-Shakarchi draw a distinction between ‘Exercises’ and ‘Problems’. The former correspond to what one will be used to from other textbooks, the latter comprise substantially more difficult guided tours of important topics in the theory of Fourier analysis, which has by now been elaborated into a high state of art in view of the fact that many of the brightest mathematicians have been exercising their ingenuity on it for well on two hundred years. For the sake of economy, this reviewer has not attempted all of the problems (to do so would render completing this textbook into a semester-long project rather than a matter of a week or two) – thus, for a long time to come, there will be plenty to refer back to on occasion and to enjoy!
  
  Conclusion: one couldn’t ask for a better introduction to Fourier analysis. The genuine lover of mathematics will delight in it and in all the fine points on which the authors dwell. Bodes well for the second, third and fourth volumes in their series of lectures on advanced undergraduate analysis from the Princeton University Press! Maybe, if one works through the present textbook – which in any event won’t take very long, it could serve as a discriminating test as to whether one is cut out to be a pure mathematician. If one finds the nice conceptual apparatus otiose and wants to press on right away to more advanced material involving harder analysis, probably one should limit the scope of one’s ambition to theoretical physics. If, on the other hand, the unhurried pace and attention to detail surrounding the basic notions strike one as appealing, perhaps it is an auspicious portent that one is favored with the gift of simplicity that makes for a good pure mathematician!
• 
  Hassan Zayour

  The authors of this book have provided some of the finest introductory books on mathematical analysis, and to whom I have become quite fond. Their style is elegant, well-organized, and distinguishably articulate. Problems are motivated in a manner that demonstrates their excellent scholarship and profound understanding of the subject, and their works would serve as excellent references for beginning and experienced mathematicians alike.
  
  Nature teaches us math like no other, and perhaps that is manifest the most in Fourier analysis, a fascinating topic that I have been perpetually enchanted with throughout my undergraduate studies. All periodic functions can be decomposed into sines and cosines, permitting us to comprehend and thoroughly analyze them. This is an example of how mathematics granted us invaluable tools to read, understand, and control some aspects of nature. I have come to appreciate this in the field of electromagnetics, where we can see this beauty in its most breathtaking forms.
• 
  Pietro

  This book is brimming with clarity and intuition. It develops basic Fourier analysis, and features *many* applications to other areas of mathematics. The proofs are elegant, the exercises terrific. It's one of the best books I have ever read.
• 
  Misrab

  Overall a great introduction to Fourier series, the transform, various types of convergence such as mean squared, Cesaro and Abel summability. What stands out are the applications to number theory e.g. finite Fourier analysis on Abelian groups, characters, primes...
  
  As always, I'm not a huge fan of the usual mathematical style of starting with the definitions and theorems, which is backwards compared to the derivation or discovery process. More context and anecdotes would be helpful for mortals. I complemented this approach by also doing Stanford's oustanding class on the Fourier transform with Brad Osgood.
  
  Overall a good book with some interesting exercises. I'll probably check out the next in the series.
• 
  Louise

  Stein is a mathematical and pedagogical genius. A thorough, yet easy to understand (for the subject matter) exposition, this is a must have text for higher study in Fourier and harmonic analysis.