  We consider a weighted form of the Poisson summation formula. We prove that
under certain decay rate conditions on the weights, there exists a unique
unitary Fourier-Poisson operator which satisfies this formula. We next find the
diagonal form of this operator, and prove that under weaker conditions on the
weights, a unique unitary operator still exists which satisfies a Poisson
summation formula in operator form. We also generalize the interplay between
the Fourier transform and derivative to those Fourier-Poisson operators.
