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Fourier Series

G. H. Hardy and W. W. Rogosinski
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Michael Berg
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There’s a somewhat hackneyed Chinese greeting that goes something like, “May you live in interesting times.” I’m told that the original meaning, or subtext, of this saying is that the one addressed might live in times of major change, generally political: flux, uncertainty, unpredictability. Manifestly we currently live in such “interesting” times, and on a parochial level particularly potent challenges are posed to the mathematical community.

In the area of mathematical pedagogy (which was an altogether alien entity in mathematical circles not all that long ago), the first two derivatives are positive and very large, which is to say that folks who claim to know something about the psychology of how mathematics is learned (and even done!) are whirring about like busy bees dripping alleged honey everywhere. The sheer quantity of workshops, online or in the real world, is daunting, with worker bees flocking to the hives with great enthusiasm and papers and books proliferating.

This frenzy to make progress has of course spread to mathematics proper, with every freshman one who can integrate the arctangent being pushed into a mathematics major (or at least a minor), with some flavor of research being the goal, and the sooner the better: every one is doing REUs, senior theses, individualized projects, and every MAA or AMS meeting these days sports hundreds of posters of student work.

With the internet providing articles on everything imaginable, the ArXiv being prominently featured, it’s an embarrassment, if not of riches, then certainly of choices. So, after the first so many siftings have taken place, and when it comes to the actual business of the few surviving genuine pure mathematicians-to-be actually learning and doing mathematics, they are likely to find themselves in dire need of guiding principles so as to make their way though mazes of choices, e.g., when it comes to texts to study. Interesting times, indeed.

But there’s light in the darkness caused by all these contemporary bee-swarms, in the form of the time-honored maxim pronounced by Niels Hendrik Abel: study the masters, not their pupils. Admittedly one should temper this judgment somewhat: acquiring the expertise and acumen required for studying a master entails an apprenticeship which involves plenty of preliminary sources. But every so often one has the great good fortune of being able to follow a preliminary course mapped out by a master: after all, even Chopin wrote many, many études.

And so did G. H. Hardy. In fact, early on in his career, and early in the 20th century, Hardy set himself the task of bringing British pure mathematics to parity with its continental counterpart, the standard having been set in the preceding two centuries largely by Germany and France. And so there flowed from Hardy’s fountain pen a number of fabulous introductory and expository works — just think of his unsurpassed A Course of Pure Mathematics and, at a higher level, one of his magna opera, An Introduction to the Theory of Numbers, written with E. M. Wright. Well, here, in the book under review, we encounter another entry in this list: the composition Hardy wrote together with W. W. Rogosinski on Fourier Series.

It’s a very short book, but dense and rich (like a diamond, one might well say). Say the authors:

“There are already a good many books on the subject [of course, nothing like what we’re faced with now (ed.)]; but we think that there is still room for one written in the modern spirit [bis.], concise enough to be included in this [Cambridge tract] series, yet full enough to serve as an introduction to Zygmund’s standard treatise … We have not written for physicists or for beginners, but for mathematicians interested primarily in the theory and with a certain foundation of knowledge.” They go on to say that the latter “can be acquired quite easily from chapters x – xii of Titchmarsh’s Theory of Functions.”

Isn’t this fabulous?: Titchmarsh to Hardy-Rogosinski to Zygmund — a well-lit path to true hard analysis with a classical flavor. (By the way, not all that long ago, I had the honor of reviewing Zygmund’s authoritative tome in this very column.)

Finally, when we dive into the text itself, we proceed from Fourier series in Hilbert space, through treatments of convergence and summability, to “general trigonometrical series,” swiftly and elegantly. It’s all scholarship of the highest order, of course, written in what is easily recognized as Hardy’s unsurpassed style, and cannot be praised too highly. God bless Dover Publications for re-issuing this book.

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

Preface; Notations
I. Generalities
II. Fourier series in Hilbert space
III. Further properties of trigonometrical Fourier series
IV. Convergence of Fourier series
V. Summability of Fourier series
VI. Applications of the theorems of Chapter V
VII. General trigonometrical series