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Publisher:

Springer Verlag

Publication Date:

2005

Number of Pages:

192

Format:

Paperback

Edition:

2

Series:

Universitext

Price:

49.95

ISBN:

0-387-22837-3

Category:

General

[Reviewed by , on ]

Michael Berg

12/1/2005

The Introduction to this wonderful book contains a clear statement of the author's own credo regarding what harmonic analysis is and should be: "...both principal incarnations of Fourier theory, the Fourier series and the Fourier transform, are special cases of a more general theory arising in the context of locally compact abelian groups." Accordingly *A First Course in Harmonic Analysis*, by Anton Dietmar, is really something of a welcome anomaly among other such first courses in the sense that Deitmar's perspective is, for lack of a better word, representation-theoretic rather than hard-analytic. His choice of material is superb and his exposition is succinct, cogent, and elegant: this is a terrific book from which to learn harmonic analysis and its connections with other parts of mathematics.

This ecumenical character of *A First Course in Harmonic Analysis* is particularly noteworthy in light of the evocative fact that harmonic analysis, or, more precisely, the theory of unitary group representations in a Hilbert space, has shown itself to be a unifying theme par excellence connecting such apparently disparate disciplines as number theory, physics, and probability theory. Naturally, in this connection the expository work of George Mackey must be mentioned, especially his books on *Unitary Group Representations in Pysics, Probability and Number Theory* and *The Scope and History of Commutative and Noncommutative Harmonic Analysis*. Deitmar's eminently readable book provides a very good introduction to the entire field Mackey addresses, and a lot more besides.

Indeed, singling out number theory for further illustration of the thesis that Deitmar's approach is hugely important, consider the eighty-year-old open problem of generalizing the Fourier-analytic proof of quadratic reciprocity, posed by Erich Hecke in his classic *Vorlesungen über die Theorie der Algebraischen Zahlen*. Hecke used Fourier series to get at a θ-functional equation tailored to yield data about Gauss sums (arising in connection with so-called θ-constants or *Thetanullwerte*). This functional equation accordingly produces a reciprocity between suitable Gauss sums which transform nicely with respect to the Legendre symbol: voilà: quadratic reciprocity à la Hecke. Gorgeous.

But Hecke went on to request a generalization of his methods to cover the case of higher reciprocity laws: a problem which is still open today. However, a major breakthrough was effected roughly forty years ago, at the hands of André Weil and Tomio Kubota. In "Sur certains groupes d'opérateurs unitaires (*Acta. Math*. 111, 1964) Weil, tackling the larger subject of C. L. Siegel's analytic theory of quadratic forms, showed that Hecke's proof is equivalent to the fact (ultimately established by means of an excursion into "abstract Fourier analysis") that the double cover of the adelic symplectic group is split on the rational points. The lynch-pin of Weil's argument is his demonstration that what is now called the Weil θ-functional is invariant under the action of the rational points (in the aforementioned adelic symplectic group) facilitated by the projective Weil representation. The latter arises in the context of the representation theory of the Heisenberg group (moved from physics to number theory!) in the presence of the theorem of Stone and Von Neumann. It was Kubota who showed somewhat later that one might get at Hecke's generalization problem by following Weil's harmonic analytic strategies applied to the case of n-fold covers of adelic SL(2) (which for n=2 is easily identified with Sp(2), the symplectic group). But these wickets are incomparably sticky and Hecke's challenge still stands.

As concerns the book under review, then, my claim is that it serves as the ideal background for getting at this part of number theory. And this point is underscored by the fact that Deitmar explicitly hits theta series (§3.7), Pontryagin duality (§7.2), representations (§9.2), the Heisenberg group (all of Chapter 12), and the θ-function (Appendix A) vis à vis the use of θ-functions and Fourier series to get functional equations.

All this having been said, I obviously recommend Deitmar's book in the strongest possible terms.

Michael Berg is Professor of Mathematics at Loyola Marymount University in California.

Fourier Series * Hilbert Spaces * The Fourier Transform * Distributions * Finite Abelian Groups * LCA groups * The Dual Group * Plancheral's Theorem * Matrix Groups * The Representations of SU(2) * The Peter-Weyl Theorem * The Heisenberg Group * The Riemann Zeta Function * Haar Integration * Bibliography * Index

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