  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.
