Abstract
The Fourier transform over a finite abelian group A is defined as a purely mathematical entity independent of its origins in physical applications. As such, the Fourier transform has been a source of many fruitful interpretations and generalizations ranging from the polynomial version of the Chinese remainder theorem to the Wedderburn structure theorem for group algebras over finite nonabelian groups.
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References
Fourier analysis over abelian groups including infinite and locally compact abelian groups can be found in many places, but one of the most readable is the Rudin text [45] which has extensive references to original sources. FFT algorithms over finite abelian groups can be found in the Tolimieri-An-Lu text [52].
Nonabelian extensions do not play any role in this work, but interested readers can consult the papers of Clausen [10] and Rockmore [44] which extend the divide-and-conquer strategy underlying most FFT algorithms to a nonabelian setting. The relationship between the Fourier transform, the Chinese remainder theorem, and the Wedderburn structure theorem is described in [10].
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© 1998 Birkhäuser Boston
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Tolimieri, R., An, M. (1998). Fourier transform over A. In: Time-Frequency Representations. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4152-2_3
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DOI: https://doi.org/10.1007/978-1-4612-4152-2_3
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-8676-9
Online ISBN: 978-1-4612-4152-2
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