Harmonic analysis is one of those exquisitely beautiful classical branches of mathematics that move one to embrace the view that mathematics is an art as opposed to a science. Of course, the border between these two categories is by no means well-defined: can there really be ugly, or non-artistic, science? Paul Dirac’s famous claim that the primary criterion for physics must be beauty or elegance, not empirical verifiability as such (the idea being that nature would not behave inelegantly) implies that an ugly physics is a wrong physics — or at least not a properly formulated one. Dirac went on to say, late in his life, after he had settled at Florida State University in 1971, that the mathematics that goes into physics must be beautiful because it was chosen, after all, by God. Then there’s the famous saying by G. H. Hardy that “there is no permanent place in the world for ugly mathematics.” It looks like Hardy would come down rather strongly on the side of mathematics being an art and only an art: this, in fact, might be proposed as one of the main thrusts of his *A Mathematician’s Apology*.

There is in any event no doubt that harmonic analysis passes the elegance test with flying colors and is remarkable, to boot, for the fact that its role beyond the borders of pure mathematics is of undeniably major proportions. Indeed, one of its pillars, the theory of Fourier series, had its genesis in Joseph Fourier’s famous 1822 work, *Théorie Analytique de la Chaleur *(i.e. “the analytic theory of heat”), square in the middle of physics.

The authors of the book under review, Muscalu and Schlag, introduce the subject accordingly. We read in their preface that “[h]armonic analysis is an old subject [that] originated with Fourier … preceded by work of Euler, Bernoulli, and others … [and the proposed] ideas were revolutionary at the time and could not be understood by means of the mathematics available to Fourier and his contemporaries.” They go on to say that “it was clear even then that the idea of representing any function as a superposition of elementary harmonics … from an arithmetic sequence of frequencies had important applications to the partial differential equations of physics that were being investigated at the time … In fact, it was precisely the desire to solve these equations that led to this bold idea in the first place.”

Once the notion of a Fourier series had taken hold, a spectacular evolution occurred, with some very famous players entering the game: Sturm and Liouville made their major contributions not long after Fourier’s work appeared, and many other redoubtable 19th century scholars worked in this area, including, for example, Dirichlet. The authors of the present book observe that “[m]any ideas that took their beginnings and motivations from Fourier series research became disciplines in their own right,” including Cantor’s set theory, Lebesgue’s measure theory, and the functional analysis of Hilbert and Banach. We begin to see the emergence of a clear path in the direction of modern developments:

harmonic analysis is therefore a vast discipline … which continues to be a vibrant research area to this day … [and] over the past 60 years Euclidean harmonic analysis, as represented by the schools associated with A. Calderón and A. Zygmund at the University of Chicago as well as those associated with C. Fefferman and E. Stein at Princeton University has been inextricably linked with … PDEs … [Additionally] the past 25 years have also seen an influx of harmonic analysis techniques to the theory of nonlinear dispersive equations such as the Schrödinger and wave equations. These developments continue to this day.

Muscalu and Schlag then go on to mention that “over the past 30 years wavelets (as well as curvelets and ringlets) have revolutionized applied harmonic analysis” and proceed to mention such things as MRIs, PETs, geophysics and oil exploration. To be sure, harmonic analysis is everywhere.

But back to pure mathematics and its pedagogy: the book we are dealing with is not a compendium or bird’s eye view of the indicated huge subject. It is in fact meant for the classroom and self-study, and proposes to get down to the mathematical nitty-gritty as a text aimed at the usual suspects, *viz. *“an advanced undergraduate or beginning graduate student.” Thus, the authors note that they are not proposing to compete with such mainstays of the subject as (what are usually just referred to as) “Stein,” “Stein and Weiss,” “Katznelson,” and “Zygmund.”

(I am a little confused about the last: the authors’ bibliography tag refers to Zygmund’s Springer Lecture Notes entry *Intégrales Singulières*, which they characterize as a “timeless encyclopedic work on Fourier series.” But *Intégrales Singulières* only weighs in at around 50 pages, is obviously narrowly focused on singular integrals, and is accordingly anything but encyclopedic. Inference: they actually mean Zygmund’s two-volume opus *Trigonometric Series*, which appears two spots lower on the bibliography list. By the way, and as a Hollywood-style plug, here’s the MAA Review of this classic work. Moreover, the aforementioned fabulous book by Katznelson has also been reviewed, by none other than our leader.)

All right, then, so Muscalu and Schlag propose to guide rightly disposed and properly prepared youngsters to harmonic analysis and they devote two relatively large volumes to the task. What do they do? Well, Volume I “could be covered by a beginning graduate student in the course of a year in independent, but guided study … culminat[ing] in some form of qualifying or ‘topic’ exam, after which the student would be expected to begin independent research.” This really means that Volume I “presents developments in harmonic analysis up to the mid to late 1980s, [after which] Vol. II picks up from there and focuses on more recent aspects … [which] can roughly be described as phase-space oriented.” Indeed, Volume II is more arcane than Volume I, but given what the texts are geared to do, i.e., get students to the point of real research as quickly as possible, that cannot be avoided.

As far as the books’ lay-out and the authors’ presentation are concerned, suffice it to say that the prose is very clear and flows well while the proofs look to be very accessible. Furthermore, the sequence of topics is clearly right on target, with, e.g., Calderón-Zygmund and Littlewood-Paley appearing in the middle of Volume I, which also contains a discussion of the uncertainty principle and closes with a discussion of the Weyl calculus, and with Volume II starting off with material on Korteweg-de Vries (a.k.a. “KdV,” of course) and then going on to such things as “paraproducts on polydisks” and “Calderón commutators and the Cauchy integral on Lipschitz curves.” Toward the end of Vol. II we encounter the very famous theme of almost everywhere convergence of Fourier series. It’s quite a line-up.

*Classical and Multilinear Harmonic Analysis* should have no problem hitting its target.

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