A family of operators satisfying a Poisson-type summation formula, and a characterization of the Fourier transform

We study to which extent the Poisson summation formula determines the Fourier transform. The answer is positive under certain technical smoothness and rate of decay conditions. This study leads to a class of unitary operators on L^2 that satisfy a weighted form of the Poisson summation formula, which we explicitly diagonalize, with eigenvalues related to associated L-functions.