Discrete Harmonic Analysis Representations, Number Theory, Expanders, and the Fourier Transform

This self-contained book introduces readers to discrete harmonic analysis with an emphasis on the Discrete Fourier Transform and the Fast Fourier Transform on finite groups and finite fields, as well as their noncommutative versions. It also features applications to number theory, graph theory, and representation theory of finite groups. Beginning with elementary material on algebra and number theory, the book then delves into advanced topics from the frontiers of current research, including spectral analysis of the DFT, spectral graph theory and expanders, representation theory of finite groups and multiplicity-free triples, Tao's uncertainty principle for cyclic groups, harmonic analysis on GL(2,Fq), and applications of the Heisenberg group to DFT and FFT. With numerous examples, figures, and over 160 exercises to aid understanding, this book will be a valuable reference for graduate students and researchers in mathematics, engineering, and computer science.

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).

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.