# Minimal Hermite-type eigenbasis of the discrete Fourier transform

## TL;DR

This paper identifies a unique optimal orthonormal eigenbasis of the discrete Fourier transform that parallels Hermite functions, satisfying an uncertainty principle, recurrence relation, and converging to Hermite functions as N grows.

## Contribution

It introduces the minimal Hermite-type eigenbasis of the DFT, demonstrating its uniqueness, optimality, and convergence properties, which were previously unexplored.

## Key findings

- Unique optimal eigenbasis of DFT identified
- Eigenvectors are analogues of Hermite functions
- Eigenvectors satisfy a three-term recurrence relation

## Abstract

There exist many ways to build an orthonormal basis of $\mathbb{R}^N$, consisting of the eigenvectors of the discrete Fourier transform (DFT). In this paper we show that there is only one such orthonormal eigenbasis of the DFT that is optimal in the sense of an appropriate uncertainty principle. Moreover, we show that these optimal eigenvectors of the DFT are direct analogues of the Hermite functions, that they also satisfy a three-term recurrence relation and that they converge to Hermite functions as $N$ increases to infinity.