Classification of Fourier summation formulas on a horizontal strip

Abstract

We extend the classification of Fourier summation formulas to the setting in which the measure $\mu$ is supported on a strip of finite width in $\mathbb{C}$. This broader framework encompasses important examples, including the Guinand--Weil explicit formulas for functions in the Selberg class, which lie beyond the scope of the previous classification. We study identities of the form \begin{align*} \sum_{n\geq0} a(\lambda_n)\varphi(\lambda_n)=\int_{\mathbb{R}} \widehat{\varphi}(t) \mathrm{d}\nu(t)+\sum_{\gamma\in A} b(\gamma)\widehat{\varphi}(\gamma), \end{align*} valid for every $\varphi\in \mathbb{C}^\infty_c(\mathbb{R})$, where $\widehat{\varphi}$ denotes the Fourier transform, $a:\{\lambda_n\}_{n\geq0}\subset\mathbb{R}\rightarrow\mathbb{C}$ is a function of finite exponential growth, $\nu$ is a Borel measure on $\mathbb{R}$ and $\eta=\sum_{\gamma\in A} b(\gamma)\delta_\gamma$ is a discrete measure supported on such a strip, with both $\nu$ and $\eta$ of polynomial growth. We prove that these summation formulas are in correspondence with almost periodic meromorphic Nevanlinna functions on the upper half-plane. More precisely, we characterize such formulas in terms of the singularities and boundary behavior of these functions, and conversely show that every function in this class gives rise to a unique Fourier summation formula of the above type.