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Generalized Functions, Volume 1: Properties and Operations

I. M. Gel'Fand and G. E. Shilov
AMS Chelsea Publishing
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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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I was blessed to receive a very eclectic university education and early mathematical formation, admittedly largely without my collusion. So it was that I learned about Shilov in a course in commutative Banach algebras given at UCLA by the late Leo Sario, with whom I had struck up a friendship based in part on our shared love of classical music and agreement that Carlo Maria Giulini was a better conductor for the Los Angeles Philharmonic than Zubin Mehta. I learned about Gel’fand in the same course, I think — well, it would certainly stand to reason, of course, but I don’t quite recall, whereas I do have particularly vivid memories of Prof. Sario talking about the Shilov boundary for a CBA. Later, and some 120 miles south at UCSD, doing modular forms with Audrey Terras, I was led in the direction of a particular flavor of representation theory and, as time went on (had I graduated with my PhD by then?), I was introduced to the legendary text by Gel’fand, Graev, and Pyatetskii-Shapiro. Thus, already at that time, during those halcyon days now many decades ago, I knew about all but one of the men who authored the books currently under review, Vilenkin being the sole exception. And it is of course particularly exciting to hold Volume 6 in my hands: nothing less than the same book on representation theory and automorphic functions just mentioned — and now it’s a bona fide, beautifully produced hard-cover re-issue of this book of which I heretofore only possessed a boot-leg (i.e. photo-copied) version. Kudos are due to the AMS and the Chelsea Series, already for just this single volume.

But what we have here in indeed a series of six books that comprises in toto a major contribution to mathematics in the wake of the introduction of one of the most important themes in modern analysis, namely, the notion of a distribution, a.k.a. a generalized function. Generalized Functions began to appear in the USSR in 1958, hot on the tracks of what had just been unleashed in France by Laurent Schwartz. Well, there is a little more to it than that, as is made clear in the Foreword to the First Russian Edition written by Gel’fand himself who makes the following observations in this Foreword:

Important to the development of the theory have been the works of Hadamard dealing with divergent integrals occurring in elementary solutions of wave equations, as well as some work by M. Riesz … The first to use generalized functions in the explicit and presently accepted form was S. L. Sobolev in 1936 … From another point of view Bochner’s theory of the Fourier transforms of functions increasing as some power of their argument can also bring one to the theory of generalized functions…

And then, the dénouement:

In 1950–1951 there appeared Laurent Schwartz’s monograph Théorie des Distributions … [and] literally within two or three years, generalized functions attained an extremely wide popularity. It is sufficient just to point out the great increase in the number of mathematical works containing the delta function…

And subsequently, a dozen pages later, as the mathematics proper begins, it all starts masterfully right from the beginning, piggy-backing off the telling mention of the delta function: “Physicists have long been using so-called singular functions, although these cannot be properly defined within the framework of classical function theory. The simplest of the singular functions is the delta function…”

With the foregoing comprising the opening salvo in the first section of the first chapter of the first volume of the series, we get, in quick succession, the notions of test functions and generalized functions. Throughout this discussion, altogether rigorous and yet having something of a very welcome informal flavor, the authors (Gel’fand and Shilov) pepper their discussion with pithy illustrations — here Dirac’s delta function is obviously the perfect choice. It is perhaps a bit amusing to note that this first chapter, titled, “Definition and simplest properties,” is on the order of 150 pages long, taking the reader all the way to Cauchy’s problem in §6.3 which culminates with the analysis of a solution of said problem for homogeneous hyperbolic PDEs involving the Herglotz-Petrovskii formulas. It is fair to say that this first chapter has in and of itself several of the qualities of an autonomous textbook: Gel’fand and Shilov deal with differentiation and integration of distributions, a lot of function theory, convolutions, and (in the aforementioned §6) DEs with constant coefficients. There are two appendices in which the authors discuss local properties and distributions depending on a parameter. (Mea culpa for going with the other terminology: it’s six of one and half a dozen of the other, of course.)

Well, the second Chapter of Book 1 deals with the hugely important subject of Fourier transforms of generalized functions, and the third chapter focuses on particular types of distributions, including, for example, those associated with quadratic functions. Beyond this — and remember, we’re only in Volume 1 — there are two Appendices, the first giving a proof of the completeness of the space of distributions, the second dealing with nothing less than the theory of distributions of one or many complex variables. As the song goes, “Who could ask for anything more?”

This first volume is really only the beginning of the story: the subsequent quintet of books in the series take it all to a higher level, as the titles indicate: Volume 2 (also 1958): “Spaces of Fundamental and Generalized Functions”; Volume 3 (still 1958): “Theory of Differential Equations,” with the Cauchy problem juiced up a bit with anabolic steroids and a new theme added:

… the apparatus of generalized functions is applied to the investigation of … the problems of determining uniqueness and correctness classes for solutions of the Cauchy problem for systems with constant (or only time-dependent) coefficients and the problem of eigenfunction expansions for self-adjoint differential operators …

Volume 4 (the date of the first Russian edition is not given): “Applications of Harmonic Analysis” (with Shilov replaced by Valenkin), and then representation theory enters the game with the last two volumes: for Volume 5 (“Integral Geometry and Representation Theory” written in 1962) Graev is added to the roster, while for Volume 6 (already mentioned above: it’s about “Representation Theory and Automorphic Functions” and goes back to at least 1962) Vilenkin goes to the bench and is replaced by Pyatetskii-Shapiro. The evident captain of the squad, I. M. Gel’fand, never leaves the court: he is the first author for all six books.

At the risk of beating a dying (but, I hope, not yet dead) horse, I must say that I can’t get enough of Volume 6. It’s really irresistible to any number theorist, myself by no means excepted: I can’t think of any real competitor as far as the treatment of this material is concerned. Of course there is also the fact that it’s déjà vu all over again for me: the book does bring back memories of my youth. But aside from that, and back to the business at hand, the text is unquestionably fantastic. The authors’ discussion of, for example, the trace formula for \(\mathrm{SL}(2,\mathbb R)\) is definitive, and then they go on to deal ever so elegantly with \(\mathrm{SL}(2)\) of locally compact topological fields. This second chapter of Volume 6 starts off with a very thorough discussion of the structure of such fields, i.e. the usual suspects, \(\mathbb{R}\), \(\mathbb C\), and of course \(\mathbb Q_p\) for \(p\) a rational prime — the latter dealt with in a very accessible and explicit manner. And they keep on going: in just this second chapter we get at irreducible representations of \(\mathrm{GL}(2,k)\) for \(k\) such a locally compact topological field (or just a local field, I guess, in this classical context), the associated discrete series, traces of irreducible representations, and Plancherel for the indicated groups. Finally in Chapter 3 of Volume 6 we get the hugely important subject of “Representations of Adèle Groups,” with Tate’s contributions (as per, e.g., his unparalleled thesis, written with Emil Artin in 1950, not all that long before this series appeared — I guess this testifies to the efficacy of the famous Gel’fand seminar) given the considerable air-play they deserve.

This classic series of six books is literally beyond praise, even given how I’ve gushed about it above.

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