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Mathematical foundation of symmetric fast Fourier transform

  • Weifeng Hou
  • Kai Jiang
  • Jingfei Tang
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Abstract

In this work, we study the fast Fourier transform (FFT) of discrete data with crystallographic symmetry, for which standard FFT algorithms incur substantial redundant computation. We establish a rigorous mathematical theory of the symmetric fast Fourier transform (SFFT) for n-dimensional crystallographic groups and propose the SFFT and symmetric inverse fast Fourier transform (SIFFT) algorithms. We derive symmetry-reduction formulas for both linear and translational symmetries and unify them into a recursive framework. We analyze the arithmetic complexity to obtain theoretical speedup factors relative to the standard FFT, and we confirm the accuracy and efficiency of the proposed methods by numerical experiments on representative three-dimensional space groups.

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Cite

BibTeXRIS
BibTeX
@misc{hou2026mathematical,
  title = {{Mathematical foundation of symmetric fast Fourier transform}},
  author = {Weifeng Hou and Kai Jiang and Jingfei Tang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11972},
  eprint = {2610.11972},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11972},
}
RIS
TY  - UNPB
TI  - Mathematical foundation of symmetric fast Fourier transform
AU  - Weifeng Hou
AU  - Kai Jiang
AU  - Jingfei Tang
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11972
UR  - https://arxiv.org/abs/2610.11972
AB  - In this work, we study the fast Fourier transform (FFT) of discrete data with crystallographic symmetry, for which standard FFT algorithms incur substantial redundant computation. We establish a rigorous mathematical theory of the symmetric fast Fourier transform (SFFT) for n-dimensional crystallographic groups and propose the SFFT and symmetric inverse fast Fourier transform (SIFFT) algorithms. We derive symmetry-reduction formulas for both linear and translational symmetries and unify them into a recursive framework. We analyze the arithmetic complexity to obtain theoretical speedup factors relative to the standard FFT, and we confirm the accuracy and efficiency of the proposed methods by numerical experiments on representative three-dimensional space groups.
ER  -