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 ℂ. Let μ=ν+η, where ν is a strongly tempered measure on ℝ, η=∑m≥1 b(γm)δγm is a strongly tempered pure point measure supported off the real line, and a:Λ→ℂ, with Λ={λn}n≥1⊂ℝ, has finite exponential growth. Under natural real-antipodal and conjugation-symmetry assumptions, we characterize summation identities of the form ∑n≥1 a(λn)φ(λn)=∫ℝ φ̂(t)dν(t)+∑m≥1 b(γm)φ̂(γm), valid for every φ∈ C∞c(ℝ), where φ̂ 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 ℂ+. The measure η encodes the poles and residues of F, while ν 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.
Related
BibTeX
@misc{vedana2026classification,
title = {{Classification of Fourier summation formulas on a horizontal strip}},
author = {Guilherme Vedana},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.10121},
eprint = {2608.10121},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.10121},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Classification of Fourier summation formulas on a horizontal strip
AU - Guilherme Vedana
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2608.10121
UR - https://arxiv.org/abs/2608.10121
AB - We classify Fourier summation identities in which the measure on the Fourier side is supported in a horizontal strip of ℂ. Let μ=ν+η, where ν is a strongly tempered measure on ℝ, η=∑_m≥1 b(γ_m)δ_γ_m is a strongly tempered pure point measure supported off the real line, and a:Λ→ℂ, with Λ={λ_n}_n≥1⊂ℝ, has finite exponential growth. Under natural real-antipodal and conjugation-symmetry assumptions, we characterize summation identities of the form ∑_n≥1 a(λ_n)φ(λ_n)=∫_ℝ φ̂(t)dν(t)+∑_m≥1 b(γ_m)φ̂(γ_m), valid for every φ∈ C^∞_c(ℝ), where φ̂ 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 ℂ^+. The measure η encodes the poles and residues of F, while ν 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.
ER -