Part I. Finite Abelian Groups and the DFT: 1. Finite Abelian groups; 2. The Fourier transform on finite Abelian groups; 3. Dirichlet's theorem on primes in arithmetic progressions; 4. Spectral analysis of the DFT and number theory; 5. The fast Fourier transform; Part II. Finite Fields and Their Characters: 6. Finite fields; 7. Character theory of finite fields; Part III. Graphs and Expanders: 8. Graphs and their products; 9. Expanders and Ramanujan graphs; Part IV. Harmonic Analysis of Finite Linear Groups: 10. Representation theory of finite groups; 11. Induced representations and Mackey theory; 12. Fourier analysis on finite affine groups and finite Heisenberg groups; 13. Hecke algebras and multiplicity-free triples; 14. Representation theory of GL(2,Fq).
A self-contained introduction to discrete harmonic analysis with an emphasis on the Discrete and Fast Fourier Transforms.
Tullio Ceccherini-Silberstein is Professor of Mathematical Analysis at Università del Sannio, Benevento. He is also an Editor of the EMS journal Groups, Geometry, and Dynamics. He has written over 90 research articles on topics ranging from functional and harmonic analysis to group theory, ergodic theory and dynamical systems, and theoretical computer sciences. He has also co-authored four monographs and four proceedings volumes. Fabio Scarabotti is Professor of Mathematical Analysis at Sapienza Università di Roma. He has written over 40 research articles on subjects ranging from harmonic analysis to group theory, combinatorics, ergodic theory and dynamical systems, and theoretical computer science. He has also co-authored three monographs. Filippo Tolli is Professor of Mathematical Analysis at Università Roma Tre, Italy. He has written over 30 research articles ranging from harmonic analysis to group theory, combinatorics, Lie groups, and partial differential equations. He has also co-authored three monographs.
'Although the roots of harmonic analysis lie in the continuous
world, in the last few decades the field has also started to play a
fundamental role in the discrete one. This book gives a panoramic
view of Discrete Harmonic Analysis - an area that touches many
branches of mathematics, such as number theory, spectral theory,
groups and their representations, and graphs. The authors open a
door for the reader taking him or her on a beautiful tour of
classical and modern mathematics All this is done in a
self-contained way that prepares the reader for cutting-edge
research.' Alex Lubotzky, Hebrew University of Jerusalem
'This book collects a number of gems in number theory and discrete
mathematics that have never been put under the same roof, as far as
I know. A distinct feature is that it puts harmonic analysis in the
foreground where most textbooks present it as ancillary results.
The authors must be complimented for their taste in the selection
of topics.' Alain Valette, Université de Neuchâtel, Switzerland
'This impressive book unites the qualities of a textbook and a
research monograph into one comprehensive text. The central theme
is the character theory of finite groups and fields, along with
various applications. It offers careful and self-contained
introductions to all required basics, which can serve for a series
of courses. At the same time, it conducts the reader through
several modern research themes and results, ranging from Tao's
uncertainty principle via expander graphs to Hecke algebras and a
detailed study of the representation theory of linear groups over
finite fields.' Wolfgang Woess, Technische Universität Graz
'The book is split up into four parts … 'Finite abelian groups and
the DFT', 'Finite fields and their characters', 'Graphs and
expanders', and 'Harmonic analysis on finite linear groups'. So
it's clear that the book covers a lot of ground, and should indeed
be of great interest to number theorists, fledgling and otherwise.
… While the book is written 'to be as self-contained as possible' ,
requiring just linear algebra up to and including the spectral
theorem, basic group and ring theory, and 'elementary number
theory', the reader is exposed to a lot of serious mathematics,
some even at or near the frontier.' Michael Berg, MAA Reviews
'The exposition of the book is kept elementary and is clear and
very readable. The selection of topics assembled in this book is
very appealing. The basics of harmonic analysis are laid out
thoroughly and in detail and at several occasions they are
complemented by non-standard applications and results which
illustrate the efficiency of harmonic analysis. In all this is a
beautiful and satisfying introduction to harmonic analysis, its
methods and applications in the discrete case.' J. Mahnkopf,
Monatshefte für Mathematik
'The book under review is a very good introduction … In a
self-contained way (it requires just elementary undergraduate
rudiments of algebra and analysis and some mathematical maturity)
it leads the reader to cutting-edge research.' Rostislav
Grigorchuk, Bulletin of the American Mathematical Society
'... a very good introduction, for researchers-in-training, to the
study of discrete harmonic analysis, its various techniques, and
its relationship to other branches of mathematics.' Mark Hunacek,
The Mathematical Gazette
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