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 $\mathbb{R}$, $\eta ={\sum }_{m\ge 1}b({\gamma }_m){\delta }_{{\gamma }_m}$ is a strongly tempered pure point measure supported off the real line, and $a:\Lambda \to \mathbb{C}$, with $\Lambda =\{{\lambda }_n{\}}_{n\ge 1}\subset \mathbb{R}$, 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(λ_n)φ(λ_n)=\int_{\mathbb{R}} \widehatφ(t)\mathrm{d}ν(t)+\sum_{m\geq1} b(γ_m)\widehatφ(γ_m), \end{align} valid for every $\phi \in C_c^{\infty }(\mathbb{R})$, where $\hat{\phi }$ 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.

Proof of Theorem 2 corrected; exposition and Selberg-class application expanded