How do you review a classic?

The most recent issue of the *Notices of the AMS** *(Vol. 59, No. 5, May 2012) contains a review by the Fefferman brothers (and a few allies), of the new book series, *Princeton Lectures in Analysis*, by E. M. Stein and Rami Shakarchi, containing the following passage by Paul Hagelstein: “As a graduate student I had studied Stein’s *Fourier Analysis on Euclidean Spaces* (coauthored with Guido Weiss), *Singular Integrals and Differentiability Properties of Functions*, and *Harmonic Analyis* in great detail and knew them to be masterpieces, the worthy sequels of Zygmund’s *Trigonometric Series*.” Thus, by transitivity, so to speak, it follows that Zygmund’s monograph is something of a grandmasterpiece, to coin a phase, and this is of course common knowledge in the analysis community. One of the aforementioned Fefferman brothers, Robert, is in fact also the author of the Foreword to the present Third Edition of *Trigonometric Series* (with Volumes I and II combined), and his marvelous essay ends with the following evaluation: “[It] is surprising … that a single person could write such an extraordinarily comprehensive and masterful presentation of such a vast field. This … is a text of historic proportion, having influenced several generations of some of the greatest analysts of the twentieth century. It holds every promise to do the same in the twenty first.”

That, by itself, should suffice to place the book under review in the front ranks of analysis texts — indeed, it is hard to imagine a competitor in the race. Even a mere fellow traveler like myself cannot help but take note of the reverence in which analysts hold this book: even when I was an undergraduate sitting in on seminars in hard analysis I very quickly found this out almost by osmosis.

Thus, this having been said, and starting off by saying that we’re dealing with a truly great book, perhaps a brief sketch of what’s found in the book’s 384 + 364 pages is appropriate.

As a product of Cambridge University Press (and the typesetting is vintage 1959, I think: the first English edition, the original appearing in 1935 in Warsaw), it has the same wonderful “feel” of such books as G. H. Hardy’s *A Course of Pure Mathematics* (than which no better introduction to true mathematics — with analysis overtones — can be imagined) and Hardy-Littlewood-Pólya on *Inequalities*; also, there’s the terrific (tiny!) book by J. C. Burkill, *The Lebesgue Integral*, which I had the pleasure to review in this column some time ago.

These books share the property of belonging to a certain generation and fitting with a certain pedagogical style, even discernible in Hardy’s *Course*: the reader is expected to be properly motivated, properly prepared, and properly willing to get his hands really, really dirty: you don’t learn to do mathematics by sitting on the sidelines.

*Trigonometric Series* is cut from the same cloth: Zygmund’s prose is terse and to the point and there’s no “padding.” One works his way through the book and then emerges with a true knowledge of, and burgeoning *Fingerspitzengefuhl* for, this part of analysis.

Zygmund takes the reader, in Part I, from a hard-core and high-level introduction to Fourier analysis (Hardy-Littlewood on p. 29) to summability questions even before we’re half-way through. At this stage, on pp. 136–137, Zygmund presents a fantastically important theme: “We [now] know that the necessary and sufficient condition for the numbers c_{v} to be the Fourier coefficients of a function in L^{2} is that the sum ∑│c_{v}│^{2} be finite. It is natural to ask whether anything so simple can be proved for the classes L^{r} where r≠2. The answer is no, and it is this fact which makes the Parseval formula and the Riesz-Fischer theorem such exceptionally powerful tools of investigation. We shall now consider criteria of a different kind involving the Cesàro or Abel means of the series considered…” How’s that for clarity and motivation?

Well, Zygmund obviously goes on to develop the indicated theme and then makes a transition to the topic of “special trigonometric series,” followed by a discussion of the question of their absolute convergence. Part I finishes with a trio of chapters on complex methods, divergence of Fourier series, and, finally, Riemann’s theory of trigonometric series. Marvelous and deep mathematics!

Part II goes on in the same (rich) vein, starting of with the important subjects of trigonometric interpolation and differentiation of series. Then earlier themes are extended — considerably — with a treatment of almost everywhere convergence and “more about complex methods.” Here the names of Marcinkiewicz, Riesz (either or both), Hardy, Littlewood, Paley, and even Edmund Landau’s “Hardy-Littlewood” appear with appropriate frequency.

Part II, and therefore the book, closed with Littlewood-Paley theory applied to Fourier series, a discussion of Fourier integrals, and finally some material on multiple Fourier series.

As already illustrated by the earlier quote from Zygmund introducing Cesàro and Abel summation, the text is eminently readable, and spurs the reader on to get into the game with no holds barred. Along these pedagogical lines, Zygmund supplies an abundance of exercises located at the end of each chapter, these being in themselves pedagogical gems: they conspire (successfully) to lead the reader to a substantial level of expertise in doing hard analysis. It is easy to see why Robert Fefferman unequivocally talks about the book’s huge influence on “several generations of some of the greatest analysts of the twentieth century.”

Additionally, in keeping with a tradition of exceptional scholarship, Zygmund provides two sections titled “Notes”, coming at the close of each Part, and containing apposite chapter-by-chapter bibliographical data, proper attributions, and editorial comments. He adds observations about open questions (at the time): the goal is, throughout, to prepare working analysts

It is easy to see why this book enjoys the majestic reputation it has: I can’t imagine a hard analyst not being enthralled by it (even today) and not wishing to learn from it.

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