{
  "abstract": "We classify Fourier summation identities in which the measure on the Fourier side is supported in a horizontal strip of $\\mathbb{C}$. Let $μ=ν+η$, where $ν$ is a strongly tempered measure on $\\mathbb{R}$, $η=\\sum_{m\\geq1} b(γ_m)δ_{γ_m}$ is a strongly tempered pure point measure supported off the real line, and $a:Λ\\rightarrow\\mathbb{C}$, with $Λ=\\{λ_n\\}_{n\\geq1}\\subset\\mathbb{R}$, 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(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 $φ\\in C^\\infty_c(\\mathbb{R})$, where $\\widehatφ$ 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 $η$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.10121",
  "author_total": 1,
  "authors": [
    {
      "name": "Guilherme Vedana"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8v2xnab515a6t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.10121",
  "fulltext_url": null,
  "id": "8v2xnab515a6t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.10121",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7202298452",
  "pdf_url": "https://arxiv.org/pdf/2608.10121",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8v2xnab515a6t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2608.10121",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.10121"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Classification of Fourier summation formulas on a horizontal strip",
  "title_text": "Classification of Fourier summation formulas on a horizontal strip",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/8v2xnab515a6t",
  "venue": "arXiv",
  "versions": []
}