A Classification of Fourier Summation Formulas and Crystalline Measures

Abstract: 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. In the Appendix we attempt to give a systematic account of the several types of Fourier summation formulas, including a new construction using eta-products, generalizing an old example of Guinand.