# Interesting Open Problem Related to Complexity of Computing the Fourier Transform and Group Theory

## TL;DR

This paper discusses an open problem in the complexity of computing the Fourier Transform, highlighting its connection to group theory and representation theory, and emphasizing the gap between known bounds and theoretical limits.

## Contribution

It introduces a natural open problem linking Fourier Transform complexity to group theory, expanding understanding of computational lower bounds.

## Key findings

- Established an $ ext{O}(n ext{log} n)$ algorithm for Fourier Transform
- Presented a lower bound of $ ext{Omega}(n ext{log} n)$ under certain conditions
- Highlighted the gap and open questions in the complexity of Fourier Transform computation

## Abstract

The Fourier Transform is one of the most important linear transformations used in science and engineering. Cooley and Tukey's Fast Fourier Transform (FFT) from 1964 is a method for computing this transformation in time $O(n\log n)$. From a lower bound perspective, relatively little is known. Ailon shows in 2013 an $\Omega(n\log n)$ bound for computing the normalized Fourier Transform assuming only unitary operations on pairs of coordinates is allowed. The goal of this document is to describe a natural open problem that arises from this work, which is related to group theory, and in particular to representation theory.