On the diagonalization of the discrete Fourier transform

The discrete Fourier transform (DFT) is an important operator which acts on the Hilbert space of complex valued functions on the ring Z/NZ. In the case where N=p is an odd prime number, we exhibit a canonical basis Φ of eigenvectors for the DFT. The transition matrix Θ from the standard basis to Φ defines a novel transform which we call the discrete oscillator transform (DOT for short). Finally, we describe a fast algorithm for computing Θ in certain cases.