Discrete Gauss-Hermite Functions and Eigenvectors of the Centered Discrete Fourier Transform

Existing approaches to furnishing a basis of eigenvectors for the discrete Fourier transform (DFT) are based upon defining tridiagonal operators that commute with the DFT. In this paper, motivated by ideas from quantum mechanics in finite dimensions, we define a symmetric matrix that commutes with the centered DFT, thereby furnishing a basis of eigenvectors for the DFT. We show that these eigenvectors in the limit converge to Gauss-Hermite (G-H) functions and that the eigenvalue spectrum of the commutor provides a very good discrete approximation to that of the continuous G-H differential operator.