When I was a graduate student in number theory in the early 1980s it transpired that in the course of working on my thesis problem (having to do with a paper by André Weil) I had occasion to read parts of Weil’s famous book, l’Intégration dans les Groupes Topologiques et ses Applications. This was a wonderful experience and counts as my first exposure to harmonic analysis, or generalized Fourier analysis, in other than the real or complex setting. My experience was not atypical and in some ways a sign of things to come: it’s doubly true today that in number theory, and even in physics, one needs to integrate in weird measure spaces, or in any case in topological contexts not usually encountered in the usual foundational courses: non-archimedean valued fields (e.g. the p-adic numbers) and rings of adèles appear on the scene, Lebesgue gives way to Haar, and that’s just the start of it — the incomparable Nachlass of Tate’s thesis, and then some.
So it is that Loomis, in the opening paragraph on the Preface to the book under review, states that this work came out of courses based on the aforementioned book by Weil taught at Harvard (presumably in the 1950s) by Mackey or himself. And, to be sure, there is recognizable resonance.
Nonetheless, Loomis’ Introduction to Harmonic Analysis is a very different book from Weil’s for a variety of reasons, the most obvious of which is pedagogical: Loomis aimed to teach this important material to an audience of graduate students. Weil, writing at an earlier time and in a different place, aimed to redress a wrong or to fill a glaring need: first taking note of then recent work by Hermann Weyl, Élie Cartan and Alfred Haar, he highlighted developments in the areas of compact and abelian groups and (generally unitary) representation theory, and then proposes “as [his book’s] objective to unite the indicated results in a systematic manner, to extend certain themes, and to clarify methods and expositions with a view to pursuing new results” [my translation]. In other words, whereas Weil was crafting tools for research per se, Loomis was preparing graduate students whose mathematical travels and adventures to date were considerably less extensive.
Beyond this Loomis notes that as the Harvard course evolved, the contemporaneous work of Gelf’and on Banach algebras began to exercise an irresistible attraction, leading to coverage of themes altogether absent from Weil’s menu. Loomis situates his coverage of topological foundations and the basics of integration (as a linear functional à la Daniell) in the first fifty pages or so of the book under review, with an introduction to Banach spaces thrown in, but after this he addresses topics that are not in Weil’s syllabus at all. Loomis quickly heads in the direction of Banach algebras as the subject of the book’s central chapters. He then proceeds to treat Haar integration at some length prior to discussing locally compact abelian groups (which, to be fair, is a major theme in Weil’s book) and finishes with compact groups and almost periodic functions. It is only in the last section of the penultimate chapter of the book that representation theory is considered, and then the book ends, still shy of two hundred pages, with “Some Further Developments.”
Thus, we are truly dealing with a discussion of abstract harmonic analysis properly so-called, and despite the book’s appearing over half a century ago, it is still a wonderfully rendered presentation of material almost every graduate student would do well to learn. It’s an old fashioned book, void of the bells and whistles commonplace today and it’s pretty much cut to the bone, but it is beautifully written, the chapters are lucidly previewed at the start of each, and the mathematics is impeccable. It’s great to have this classic available in a Dover printing.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.