{
  "abstract": "This note grew out of two questions arising in different settings. The first concerns recognizing holomorphic functions among continuous functions on an annulus. The second concerns what self-reproducing subsets can tell us about semigroups of functions arising in analysis. Our analytic starting point is a variation on the classical relation between holomorphy and invariance of contour integrals. It identifies the holomorphic functions as the largest class containing the constant function 1 for which a contour convolution is independent of the chosen curve for every pair of functions. On this class, convolution amounts to multiplying Laurent coefficients. We then study families $B$ satisfying $B\\diamond B=B$. We show that expressing a function as a convolution product of $m$ holomorphic functions corresponds precisely to its extension to a wider annulus determined by $m$. This identifies the largest self-reproducing family as the functions extending holomorphically to the punctured plane. We also prove that uniform boundedness on a fixed subannulus surrounding the unit circle forces all members of such a family to have Laurent expansions supported on one common finite set. A general semigroup theorem describes the minimal self-reproducing families.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19197",
  "author_total": 1,
  "authors": [
    {
      "name": "Gabriel Ben-Simon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b1048axgjfn24.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19197",
  "fulltext_url": null,
  "id": "b1048axgjfn24",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19197",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7222255305",
  "pdf_url": "https://arxiv.org/pdf/2610.19197",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b1048axgjfn24.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19197",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19197"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Analytic Properties of Self-Reproducing Families under Contour Convolution",
  "title_text": "Analytic Properties of Self-Reproducing Families under Contour Convolution",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/b1048axgjfn24",
  "venue": "arXiv",
  "versions": []
}