Fourier Integrals in Classical Analysis

This semester, for the first time in quite a while, I am teaching introductory calculus, not for just anyone but the serious kind, for “for scientists and engineers,” which gives me license to do some real mathematics for the children, if only very gently. For other calculus clients, well, let’s just say there are proofs and then there are proofs. Anyhow, what is forever a source of amazement to me, even now, almost 50 years since I first encountered these notions, is the malleability of the ideas as the base of the edifice of analysis. Back in the late 1970s, the prominent topologist Bob Edwards, lecturing to a topology class I was in, characterized the derivative as “the best linear approximation” to a suitable geometric object, e.g. a manifold, and it is a wonderful pedagogical device to stress this pithy characterization throughout the trajectory of this notion in different courses. Then there is that miraculous connection between the derivative and the integral, as per the Fundamental Theorem of Calculus. It is still striking to me to reflect on how the evolving slope of a curve in the plane dictates the accumulation of subtended area underneath: what a fantastic way to look at what’s happening.

But there is so much more: I never pass up the chance to stress to my students that, indeed, the integral is more important than the derivative — a purposely controversial claim I make as much to shock them and tweak their budding intuitions, as to foretell what comes later, namely the gigantic evolutionary sweep of the notion of the integral as one travels through higher mathematics and even physics. We go, after all, from Riemann to Stieltjes to Lebesgue, and then to Haar and … where? Well, even to Feynman, modulo a lot of problems that we mathematicians have to take a lot more seriously than Feynman’s physicists do. As a number theorist I have dealt with Haar integrals since pretty early on, and I’m guardedly happy about what has happened to me lately, namely, that my research has brought me face-to-face with Feynman’s path integrals, the life’s blood of so much of quantum field theory. On the way, I have been exposed to all sorts of marvels in the integral family, including Fourier integrals (already in the game early on in analytic number theory) and (for me coming rather later) oscillatory integrals. The latter are in fact also of particular interest at least to certain physicists.

The book under review is manifestly concerned with Fourier integrals, and very early on it starts in on oscillatory integrals, too, but another primary feature Sogge covers is microlocal analysis. In my own experience, I first came across this material obliquely, in the fabulous book Sheaves on Manifolds, by Masaki Kashiwara (one of the foremost modern players in the subject of microlocal analysis) and Pierre Schapira. Sogge notes in his preface to the present (second) edition that “[his] goal for the first as well as the current edition was to provide a self-contained but by no means exhaustive introduction to harmonic and microlocal analysis.”

Now, also in light of my earlier remarks, we all have a pretty good idea of what harmonic analysis entails, what with its close relationship with Fourier analysis and the prominence in the subject of the eigenvalue problem in a very general form, but what about microlocal analysis? Well, the following characterization is lifted from the Introduction to Microlocal Analysis by MIT’s Richard Melrose (cf. also the Wikipedia entry that refers to it): “Microlocal analysis is a geometric theory of distributions, or a theory of geometric distributions. Rather than study general distributions (which are like general continuous functions but worse) we consider more specific types of distributions which actually arise in the study of differential and integral equations.” And this orientation is on display in the book under review.

So, let’s get to the book itself, then. After an introductory chapter covering background material, including stuff on fractional integration and the cotangent bundle (needed particularly for microlocal analysis) and also oscillatory integrals, the all-important subject of stationary phase is covered, and one might bear in mind the role this methods also plays in Feynman integrals. Then it’s on to the integral operators in play, including pseudo-differential ones (and others like non-homogeneous oscillatory integral ones), and finally Fourier integral operators about half-way through the book. And the point of it all is of course what’s generally called hard analysis. The book’s later chapters deal with such things as propagation of singularities, local smoothing of Fourier integral operators, and e.g. the Kakeya Problem. It turns out that, indeed, this is the famous “needle problem,” asking for the smallest planar set (small in the sense of Lebesgue) in which a unit line segment (a “needle”) can turn in all directions. Sogge mentions (p. 268) the famous result of Besicovitch going back to 1919, to the effect that there are compact sets of planar Lebesgue measure zero that do the job. Sogge then goes on to look at Kakeya (or Besicovitch) sets from a slightly different perspective. Say that a set of measure \(0\) in \(\mathbb{R}^n\) is a Kakeya set iff it contains a unit line segment in every direction (as determined by \(S^{n-1}\)); the question is whether such sets always have dimension \(n\). On p. 274 we find the result by R. O. Davies answering this question in the affirmative in the case \(n=2\). But then, on p.275, we read that “in higher dimensions it remains a major open problem.” Very nice. Sogge says a lot more, but for that see the book.

Fourier Integrals and Classical Analysis is an excellent book on a beautiful subject seeing a lot of high-level activity. Sogge notes that the book evolved out of his 1991 UCLA lecture notes, and this indicates the level of preparation expected from the reader: that of a serious advanced graduate student in analysis, or even a beginning licensed analyst, looking to do work in this area. But a lot of advantage can be gained even by fellow travelers, all modulo enough mathematical maturity, training, and Sitzfleisch.

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