  We define the quadratic algebra su(2)_{\alpha} which is a one-parameter
deformation of the Lie algebra su(2) extended by a parity operator. The
odd-dimensional representations of su(2) (with representation label j, a
positive integer) can be extended to representations of su(2)_{\alpha}. We
investigate a model of the finite one-dimensional harmonic oscillator based
upon this algebra su(2)_{\alpha}. It turns out that in this model the spectrum
of the position and momentum operator can be computed explicitly, and that the
corresponding (discrete) wavefunctions can be determined in terms of Hahn
polynomials. The operation mapping position wavefunctions into momentum
wavefunctions is studied, and this so-called discrete Hahn-Fourier transform is
computed explicitly. The matrix of this discrete Hahn-Fourier transform has
many interesting properties, similar to those of the traditional discrete
Fourier transform.
