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

The problem of furnishing an orthogonal basis of eigenvectors for the discrete Fourier transform (DFT) is fundamental to signal processing. Recent developments in the area of discrete fractional Fourier analysis also rely upon the ability to furnish a basis of eigenvectors for the DFT or its centralized version. However, none of the existing approaches are able to furnish a commuting matrix where both the eigenvalue spectrum and the eigenvectors are a close match to corresponding properties of the continuous differential Gauss-Hermite operator. Furthermore, any linear combination of commuting matrices produced by existing approaches also commutes with the DFT, thereby bringing up the question of uniqueness. In this paper, inspired by concepts from quantum mechanics in finite dimensions, we present an approach that furnishes a basis of orthogonal eigenvectors for both versions of the DFT. This approach also furnishes a commuting matrix whose eigenvalue spectrum is a very close approximation to that of the Gauss--Hermite differential operator and consequently a framework for a unique definition of the discrete Gauss--Hermite operato