The discrete Fourier transform: A canonical basis of eigenfunctions

TL;DR

This paper introduces a canonical eigenbasis for the discrete Fourier transform when N is an odd prime, and proposes a new transform called the discrete oscillator transform (DOT) with an efficient computation algorithm.

Contribution

It presents a canonical eigenbasis for the DFT at prime N and introduces the discrete oscillator transform with a fast computation method.

Findings

Canonical eigenbasis for DFT at prime N

Definition of the discrete oscillator transform (DOT)

Fast algorithm for computing DOT in certain cases

Abstract

The discrete Fourier transform (DFT) is an important operator which acts on the Hilbert space of complex valued functions on the ring Z/NZ. In the case where N=p is an odd prime number, we exhibit a canonical basis of eigenvectors for the 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). Finally, we describe a fast algorithm for computing the DOT in certain cases.