Intertwining relations between the Fourier transform and discrete Fourier transform, the related functional identities and beyond

摘要

Starting from the spectral decomposition of the matrix or operator roots of unity we derive in a very simple way the connection between Gauss sums and spectral multiplicities of the discrete Fourier transform (DFT), also known as Schur matrix Φ(n) or as quantum Fourier transform. Next we propose simple explicit construction of the real orthogonal matrices O_n diagonalizing Φ(n). We establish different intertwining relations between Fourier transform (FT) and DFT coming from the knowledge of the Gauss sums and Poisson summation. Finally, we present a way to generate the eigenvectors of the DFT involving the eigenfunctions of the FT or any absolutely convergent series. This gives us a source for generating various linear and nonlinear functional identities: in particular, some theta functional identities.