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Browse the figure galleryThe DAFT book · Revised English edition
The Discrete Algebra of the Fourier Transform
The algebra behind Fourier analysis, from finite groups and fast algorithms to error-correcting codes and representations. Mathematical ideas developed through proofs, exercises, and working Python examples.
PDF · 206 pages · 4.2 MB · September 2026
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From characters to computation
Read a chapter in the PDF, or select an illustration to explore its mathematics. Each chapter ends with its exercise solutions; two Python appendices complete the book.
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01Fourier analysis on finite groups
Characters, duality, orthogonality, convolution, and random walks.
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02Applications of Fourier duality
Gauss sums, quadratic reciprocity, Walsh transforms, and Poisson summation.
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03The discrete Fourier transform
FFT factorizations, multidimensional transforms, interpolation, and spectral structure.
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04Applications of the discrete Fourier transform
Filtering, geometric descriptors, differential equations, and fast products.
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05Transforms related to the DFT
Hartley transforms, the z-transform, chirp algorithms, and scaled frequencies.
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06Fourier transforms over finite fields and rings
Exact transforms, Hamming and BCH codes, decoding, and Boolean functions.
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07Linear representations of finite groups
Irreducibility, characters, matrix-valued Fourier analysis, and finite-field wavelets.
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Cube symmetries
08Applications of linear representations
Character tables, cube symmetries, ranking data, quaternions, and invariant polynomials.
An English adaptation of the French original
This book is an adapted English translation of Gabriel Peyré's L'algèbre discrète de la transformée de Fourier, published by Ellipses in 2004, ISBN 9782729818678.
The September 2026 edition develops the proofs and exposition with 68 figures and Python examples throughout. It assumes undergraduate linear algebra and basic familiarity with groups and complex numbers.
The previous English edition and its original programs remain available in the legacy PDF and archived sources.
Figures and Python
Python examples
The book's numerical and symbolic programs, with shared helpers and installation instructions. Use them to explore transforms, finite groups, interpolation, and error-correcting codes.
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