Fourier Analysis on Finite Abelian Groups ends with a discussion of on of the staples of number theory, namely, quadratic Gauß sums and the law of quadratic reciprocity. This is unquestionably one of the most beautiful themes in classical (or any) mathematics and a favored theme of Gauß, who gave six (some say seven) proofs of quadratic reciprocity in his lifetime. And this theme has stayed fecund through the succeeding centuries, of course, so it is fitting that it should be represented in an entry in Birkhäuser’s “Applied and Numerical Harmonic Analysis” series. Indeed, there is a real sense in which number theory is applied mathematics, given that its practice requires a kit full of diverse and even disparate tools. Again, Gauß’ half-dozen proofs aimed at disclosing a corresponding number of connections (at least) with other parts of mathematics: a large tool kit is necessary a priori. The proof of quadratic reciprocity closing the book under review is fittingly classed under applied harmonic analysis: that is precisely what it is.
Additionally, it is a burgeoning motif in modern mathematics that number theory and algebraic geometry on the one hand, and quantum physics and cosmology on the other, should cross-fertilize each other. A marvelous example of this is the recent work by Witten or by Kontsevich; then there is the work by Tom Bridgeland and Mike Douglas where physics’ D-branes are made to give rise to avant garde stability constructs on derived categories of coherent sheaves in algebraic geometry. Against this back ground the penultimate chapter of Fourier Analysis on Finite Abelian Groups, “The quantum Fourier transform,” may be mentioned as an (admittedly very early) harbinger of things to come for the pupil disposed to meditate on this deep synergy between “the mathematics God chose for physics,” to paraphrase P. A. M. Dirac, and Gauß’ famous Queen of the Sciences. (Would the great man disagree with the polemic that mathematics is an art, really?) In any case, this nice chapter is a brief but evocative discussion of what happened to the Fourier transform in the hands of, yes, Dirac (bras, kets, and all that). After all, it all started with quantum mechanics and relativity (viz. Hilbert, Weyl, Dirac, von Neumann, &c.) — “it” being the aforementioned synergy between physics and mathematics, of course.
Thus, the book under review covers, qua orientation, a pretty broad spectrum, if I may be forgiven an egregious pun, and the mathematics developed in its pages is well chosen for this purpose. The author, Bao Luong, targets well-prepared upper-division students and certain “outsiders” (scientists and engineers) and has taken pains to make his presentation accessible. At the same time, he hits pretty hard: linear functionals and dual spaces on p. 26, character theory on p. 33, “Is the Fourier transform a self-adjoint operator” on p. 65, uncertainty on p. 76, tensor decompositions of FT’s on p. 84, the spectral theorem on p. 104, and the ergodic theorem on p. 109.
The pace notwithstanding, this compact book is indeed very readable; however, the instructor (of, it is to be hoped, a strong and motivated class) had better be quite comfortable with the indicated material: despite the fact that it’s all there in an orbit of 150 pages (Fourier Analysis on Finite Abelian Groups is presented as self-contained), a good deal of additional motivation as well as further elaboration of certain arguments and topics seem strongly indicated, at least for gifted rookies. À propos gifted rookies, there are fifty-six exercises scattered throughout the text, generally quite sporty.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.