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Fourier analysis on finite groups and applications. (English) Zbl 0928.43001
London Mathematical Society Student Texts. 43. Cambridge: Cambridge University Press. x, 442 p. £18.95, $29.95 pbk; £50.00,$ 74.95 hbk (1999).

In the heading to Chapter III of her previous two-volume book “Harmonic Analysis on Symmetric Spaces and Applications, Part I (Berlin 1985; Zbl 0574.10029), Part II (Berlin 1988; Zbl 0668.10033)” Audrey Terras cites the question “But why then, this mystical set-up of putting the definition before the proof?”. It is a very good question – after all it was posed by Imre Lakatos – and one which should be confronted by those teaching mathematics at a higher level. The author attempts in this book, as she did in the previous ones, to present mathematics in its rich, often messy, vitality. Many books at this level are written under the influence of the axiomatic method; the motivation and broader ideas are left to university teachers to communicate to the students. This limits the number of people who can profit from them. The strategy here is an attempt to be more democratic in that the student is presented with much more of the broader context of the material. This approach is not without its costs. First of all a great deal of the material is in the form of exercises – with hints; in his review of the previous books J. Elstrodt (Zbl 0574.10029) wrote that a more descriptive title might be “Teach yourself harmonic analysis on symmetric spaces by examples”; this book could be described in much the same way. It therefore assumes a fair degree of technical competence on the part of the student. Secondly, the author is very widely read and there is a huge bibliography and numerous references in the text. The author is very broadly-read; she has included a huge bibliography and the text contains many references. Perhaps the book should have been produced in hypertext; for a student for whom this entire area of mathematics is new and who takes these references seriously it will be fairly daunting, at least until she or he learns to treat them as red herrings to be ignored. It should also be added that not many universities will be able to supply all the literature that is referred to.

The material covered in the first part of the book is the theory of representations of finite abelian groups and their manifold applications. The second part is devoted to the theory of non-abelian finite groups and applications to certain special groups related to abelian ones, such as finite Heisenberg groups and $GL\left(2,{𝐅}_{q}\right)$. In other words these are the examples that are the analogues of the ideas met in the theory of automorphic forms. The choice of material is modern and many of the applications are inspired by questions in communication theory (expansiveness of graphs, Ramanujan graphs and other questions involving isoperimetric properties of graphs, error-correcting codes, “Buckyballs”, etc.). Moreover, it is not easy to find this material in a usable form for teaching and this book will be very useful for this reason alone. The author has illustrated many of the results by computer generated graphics; in doing so she has stressed the usefulness of Mathematica and MathLab; it is worth noting that such results can be obtained without too much trouble on simpler and cheaper systems.

In the first instance this book has been written for students and it should fire the imagination of some. It may, however, irritate others, but those who it does inspire will be very well prepared for further work. For the teacher it is a rich source of ideas and examples, as well as representing a challenge to find better and more stimulating methods of teaching mathematics, methods which are suitable for the present generation of students.

##### MSC:
 43-02 Research monographs (abstract harmonic analysis) 65T50 Discrete and fast Fourier transforms (numerical methods) 11F72 Spectral theory; Selberg trace formula 05C25 Graphs and abstract algebra