Application of the Weil representation: diagonalization of the discrete Fourier transform

TL;DR

This paper explores how the Weil representation can be used to diagonalize the discrete Fourier transform, introducing a new basis and a fast algorithm for the resulting discrete oscillator transform.

Contribution

It presents a novel application of the Weil representation to construct a canonical eigenbasis for the DFT and introduces a new fast computation algorithm for the DOT.

Findings

Constructed a canonical eigenbasis for the DFT using Weil representation.

Defined a new transform called the discrete oscillator transform (DOT).

Developed a fast algorithm for computing the DOT in specific cases.

Abstract

We survey a new application of the Weil representation to construct a canonical basis of eigenvectors for the discrete Fourier transform (DFT). The transition matrix from the standard basis to the canonical basis defines a novel transform which we call the discrete oscillator transform (DOT for short). In addition, we describe a fast algorithm for computing the DOT in certain cases.