A discrete Fourier transform on a q-linear grid is presented. Several of its analytic properties are discussed and compared with the continuum situation. We recognize that a special invariant of the related Fourier operator is closely connected to discretizations of the Hermite functions. This result is similar to the continuum scenario and reveals a key role of the q-Fourier transform to the understanding of difference equations. The continuum limit q1 in the sense of strong L2-convergence is investigated for the derived q-Fourier invariant generalization of the Gauss curve. Applications to Schrödinger difference equations are briefly mentioned.