Intertwining relations between the Fourier transform and discrete Fourier transform, the related functional identities and beyond
Vladimir Borisovich Matveev · Inverse Problems · 2001
Starting from the spectral decomposition for 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.