A classification of Fourier summation formulas and crystalline measures
Felipe Gonçalves
TL;DR
This work delivers a complete classification of Fourier summation formulas (FS-pairs) and crystalline measures with quadratic decay, by developing a unified framework anchored in almost periodic analysis, Hermite–Biehler functions, de Branges spaces, and Poisson representations. A central contribution is a bijective correspondence between FS-pairs with degree at most two and holomorphic upper-half-plane data encoded by functions f whose exponentials exhibit almost periodicity and bounded-type behavior; this yields explicit limit formulas and spectral constraints, and conversely constructs FS-pairs from such f. The authors embed FS-pairs within de Branges space theory, providing a Hilbert-space interpretation, kernel representations, and phase-zeros interplay that recovers and extends prior constructions (Kurasov–Sarnak, Olevskii–Ulanovskii, Guinand). They also present broad families of examples, including UDPS, RRTP, and particularly eta-quotient-based self-dual crystalline measures, linking to classical Poisson summation and prime-sum formulas. Overall, the paper advances the structural understanding of crystalline measures, offers new methods for generating self-dual examples, and connects Fourier-analytic classifications to deep operator-theoretic and number-theoretic frameworks. $
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. Moreover, we give a new construction using eta-quotients, generalizing an old example of Guinand.