  The mathematical description of the quantum harmonic oscillator is
essentially based on the Gaussian function. In the case of a quantum oscillator
with finite-dimensional Hilbert space, the position space consists in a finite
number of points, and the ground state can be regarded as a finite version of
the Gaussian function. Conversely, each finite Gaussian can be used in order to
define some finite oscillators. We investigate certain finite Gaussians as
concern their Fourier transform, Wigner function and associated oscillators.
