A classification of Fourier summation formulas and crystalline measures

We completely classify Fourier summation formulas, and in particular, all crystalline measures with quadratic decay. Our classification employs techniques from almost periodic functions, Hermite-Biehler functions, de Branges spaces and Poisson representation. We show how our classification generalizes recent results of Kurasov \& Sarnak and Olevskii \& Ulanovskii. As an application, we give a new classification result for nonnegative measures with uniformly discrete support that are bounded away from zero on their support. Moreover, we give a new construction using eta-quotients, generalizing an old example of Guinand.