  We construct a new model of the quantum oscillator, whose energy spectrum is
equally-spaced and lower-bounded, whereas the spectra of position and momentum
are a denumerable non-degenerate set of points in [-1,1] that depends on the
deformation parameter q from (0,1). We provide its explicit wavefunctions, both
in position and momentum representations, in terms of the discrete q-Hermite
polynomials. We build a Hilbert space with a unique measure, where an analogue
of the fractional Fourier transform is defined in order to govern the time
evolution of this discrete oscillator. In the limit q to 1, one recovers the
ordinary quantum harmonic oscillator.
