  The ``position'' and ``momentum'' operators for the q-deformed oscillator
with q being a root of unity are proved to have discrete eigenvalues which are
roots of deformed Hermite polynomials. The Fourier transform connecting the
``position'' and ``momentum'' representations is also found
  The phase space of this oscillator has a lattice structure, which is a
non-uniformly distributed grid. Non-equidistant lattice structures also occur
in the cases of the truncated harmonic oscillator and of the q-deformed
parafermionic oscillator, while the parafermionic oscillator corresponds to a
uniformly distributed grid.
