Hermite Functions play a fundamental role in the theory of Fourier Transform as eigenvectors of the Fourier Transform, a basis of $L^2$ functions in $\mathbb R$ and a representation of the Lie algebra $h(1)$. We exhibit that, with an appropriate discretization and periodization, they cover the same role for Fourier Series and their inverse, the Discrete Time Fourier Transform.

<p>Discretized and Periodized Hermite Functions are indeed:
</p><p>1) Related by Fourier Series and its inverse.
</p><p>2) Bases of the spaces $L^2({\mathbb C})$ and $L^2({\mathbb Z})$ where Fourier Series and their inverse act.
</p><p>3) Representations of the same algebra $h(1)$.</p>