# Fast generalized DFTs for all finite groups

## TL;DR

This paper introduces an efficient arithmetic algorithm for computing generalized Discrete Fourier Transforms over any finite group, significantly improving computational complexity by leveraging matrix multiplication exponents.

## Contribution

It provides the first general algorithm for all finite groups with complexity tied to the matrix multiplication exponent, advancing computational harmonic analysis.

## Key findings

- Achieves $O(|G|^{rac{ ext{omega}}{2} + ext{epsilon}})$ complexity for all finite groups.
- Extends the applicability of fast Fourier transform techniques to arbitrary finite groups.
- Bridges group theory and matrix multiplication complexity for efficient Fourier computations.

## Abstract

For any finite group $G$, we give an arithmetic algorithm to compute generalized Discrete Fourier Transforms (DFTs) with respect to $G$, using $O(|G|^{\omega/2 + \epsilon})$ operations, for any $\epsilon > 0$. Here, $\omega$ is the exponent of matrix multiplication.

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Source: https://tomesphere.com/paper/1901.02536