Discrete fractional Fourier transform: Vandermonde approach

Based on the definition of the continuous Fourier transform in terms of the number operator of the quantum harmonic oscillator and in the corresponding definition of the continuous fractional Fourier transform, we have obtained the discrete fractional Fourier transform from the discrete Fourier transform in a completely analogous manner. To achieve this, we have used a very simple method based on Vandermonde matrices to obtain rational and irrational powers of the discrete Fourier transform. An advantage of our proposal is that it does not use the eigenvectors of the discrete Fourier transform matrix, for which there is not a simple analytical general formula and which are not unique.