Explicit Hermite-type eigenvectors of the discrete Fourier transform

TL;DR

This paper constructs explicit eigenvectors of the discrete Fourier transform, orthogonalizes them, and shows the first eight approximate Hermite functions, advancing understanding of their relationship.

Contribution

It provides an explicit basis of eigenvectors for the discrete Fourier transform and demonstrates convergence of the first eight to Hermite functions.

Findings

First eight eigenvectors converge to Hermite functions

Explicit basis of eigenvectors constructed

Orthogonal eigenbasis obtained via Gram-Schmidt

Abstract

The search for a canonical set of eigenvectors of the discrete Fourier transform has been ongoing for more than three decades. The goal is to find an orthogonal basis of eigenvectors which would approximate Hermite functions -- the eigenfunctions of the continuous Fourier transform. This eigenbasis should also have some degree of analytical tractability and should allow for efficient numerical computations. In this paper we provide a partial solution to these problems. First, we construct an explicit basis of (non-orthogonal) eigenvectors of the discrete Fourier transform, thus extending the results of [7]. Applying the Gramm-Schmidt orthogonalization procedure we obtain an orthogonal eigenbasis of the discrete Fourier transform. We prove that the first eight eigenvectors converge to the corresponding Hermite functions, and we conjecture that this convergence result remains true for all eigenvectors.