Fourier Analysis, Part I – Theory

Surely everyone loves Fourier analysis: it’s beautiful, it’s everywhere, it’s extremely important, and it’s extremely useful. Not only do the physicists use it all the time, particularly when sailing their flagship of quantum mechanics (where it’s given a steroid injection to make it into functional analysis), but it was in fact discovered by the fellow after whom it’s named in order to deal with physical phenomena like the theory of heat (cf. his Théorie Analytique de la Chaleur) and of course the vibrating string.

When I was young and stupid, and in my laziness tried to avoid hard analysis, I pretty much identified Fourier analysis with the study of trigonometric series, and thereby justified playing it down: I really couldn’t see much of a connection to the number theory I was heading toward. I had no idea that as I matured and got deeper and deeper into number theory, behold, there it was, flamboyantly entering the scene later in the play. It started to come home to me when I began learning about modular forms, whose Fourier expansions are certainly beautiful, everywhere, important, and useful.

In due course I read Hecke’s Vorlesungen über die Theorie der Algebraischen Zahlen, thinking that in going at algebraic number theory I wouldn’t see Fourier analysis — not that I was still trying to avoid it, but it didn’t seem likely to appear, given the competing treatments I’d seen at that point, e.g. Lang’s Algebraic Number Theory or Samuel’s Théorie Algébrique des Nombres; I’d also read part of Serre’s Cours d’Arithmétique and Corps Locaux, all ultimately dominated by commutative algebra. But then there was Hecke. The crowning jewel of his Vorlesungen is a proof of the relative quadratic reciprocity law in Gauss-Euler form (i.e. quadratic reciprocity for arbitrary global fields expressed in terms of generalized Legendre symbols) by means of a fabulous manipulation of the functional equation of what are now called Hecke \(\vartheta\)-functions, their Fourier series front and center. This is very, very deep stuff, given the fact that Hecke \(\vartheta\)-functions are half-integral weight modular forms and their definition depends on nothing less than the heat kernel. We see here the striking kinship, so to speak, between number theory and physics — I recall my undergraduate professor, V. S. Varadarajan, mentioning to me that number theory and physics are two sides of the same coin. So, in contrast to the shortsighted laziness of my youth I now stand corrected: it’s like cognac, perhaps, in that it’s really best appreciated once you’ve been around for a while.

Very well, then, I guess my point is that no one really should really need to be sold on how important it is for essentially all mathematicians to learn Fourier analysis, which brings me to the book under review. And, in a word, it’s fabulous. Harking back once again to my school days long ago, in a rare moment of repentance for my skirting analysis to the extent that I had (yes, not just Fourier analysis — young and stupid, indeed), I asked Steve Krantz for a recommendation for me to try to make up some of the lost ground, and he pointed me in the direction of Pólya-Szegő, Aufgaben und Lehrsätze aus der Analysis (or Problems and Theorems in Analysis), and I must say the present book by Constantin reminds me of that classic, if necessarily only to a limited extent. Pólya- Szegő is a milestone in mathematical pedagogy (in the old-fashioned, well-defined meaning of that word) and is very broad in its scope; in this connection see the review by none other than Harley Flanders. Constantin’s book has a far narrower scope, of course, and there are other pedagogical differences. But the principal similarity is profound: problems, problems, problems, and the solutions are provided for the student’s great benefit. Constantin structures his exercise sets beautifully, I think: they are abundant and long, covering a spectrum of levels of difficulty; each set is followed immediately by a section of hints (in one-one correspondence); finally the hints sections are followed by very detailed and well-written solutions (also bijectively). Can there be any clearer homage to the maxim that to learn mathematics one has to get one’s hands really dirty?

To boot, attention to detail is ubiquitous: it’s everywhere in Constantin’s presentation of proofs and arguments, as well as examples, all throughout the narrative itself. The entire presentation is very much to the point and the student who works through this book will come out knowing some real mathematics very well.

Here are the topics covered, in order. In a wonderful introductory chapter Constantin presents the aforementioned motivations regarding the theory of heat and the vibrating string, and even presents the absolutely irresistible proof of the irrationality of \(\pi\). After that we get the usual suspects — well, old friends, I guess: Lebesque integration, functional analysis, convergence of Fourier series, the Fourier transform, Fourier analysis in higher dimensions, and then “some advanced topics” (and we meet Paley-Wiener and Hardy).

Two final points. Constantin adds a clincher at the end of the book, viz. an appendix titled “Historical Notes,” in which he presents a descriptive line or so for a host of mathematicians associated with Fourier analysis. I think this is a good touch. And lastly I want to note that this excellent book is subtitled, “Part 1 – Theory.” Draw your own (happy) conclusion. 

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.