Classification of Fourier summation formulas on a horizontal strip

Abstract

We classify Fourier summation identities in which the measure on the Fourier side is supported in a horizontal strip of $\mathbb{C}$. Let $\mu=\nu+\eta$, where $\nu$ is a strongly tempered measure on $\mathbb{R}$, $\eta=\sum_{m\geq1} b(\gamma_m)\delta_{\gamma_m}$ is a strongly tempered pure point measure supported off the real line, and $a:\Lambda\rightarrow\mathbb{C}$, with $\Lambda=\{\lambda_n\}_{n\geq1}\subset\mathbb{R}$, has finite exponential growth. Under natural real-antipodal and conjugation-symmetry assumptions, we characterize summation identities of the form \begin{align} \sum_{n\geq1} a(\lambda_n)\varphi(\lambda_n)=\int_{\mathbb{R}} \widehat{\varphi}(t)\mathrm{d}\nu(t)+\sum_{m\geq1} b(\gamma_m)\widehat{\varphi}(\gamma_m), \end{align} valid for every $\varphi\in C^\infty_c(\mathbb{R})$, where $\widehat{\varphi}$ denotes the Fourier transform. We prove that each such identity determines a unique generating function $F$ that is holomorphic and almost periodic in the half-plane above the strip and admits a meromorphic continuation to $\mathbb{C}^+$. The measure $\eta$ encodes the poles and residues of $F$, while $\nu$ describes the boundary behavior of its regular part through a generalized Nevanlinna representation, and $a$ determines its Fourier coefficients. Conversely, every function in the corresponding meromorphic class whose Fourier coefficients satisfy a local summability condition determines a unique summation identity of this form. The proof combines a strip version of the Bridge Lemma with a Cauchy-transform argument that accounts for the off-real poles. As an application, we show that the Guinand-Weil explicit formula for every member of the Selberg class, including non-self-dual members, fits into our framework, and we identify its associated generating function.