Discrete Harmonic Analysis

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

• 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).