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arXiv · 2312.11185

A classification of Fourier summation formulas and crystalline measures

Felipe Gonçalves
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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.

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Felipe Gonçalves. 2023-12-18. A classification of Fourier summation formulas and crystalline measures. https://arxiv.org/abs/2312.11185

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