{
  "abstract": "Let $\\mathcal{H}(b)$ be the de Branges-Rovnyak space associated to a non-extreme point $b$ of the unit ball of $H^\\infty$, and let $ϕ=b/a$, where $a$ is the Pythagorean mate of $b$. It is known that, if $f$ is a function holomorphic on a neighbourhood of the closed unit disk, then it belongs to $\\mathcal{H}(b)$, and its norm in $\\mathcal{H}(b)$ can be expressed in terms of the Taylor coefficients of $f$ and $ϕ$ via the formula \\[ \\|f\\|_{\\mathcal{H}(b)}^2=\\sum_{m\\ge0}|\\hat{f}(m)|^2 +\\sum_{m\\ge0}\\Bigl|\\sum_{n\\ge0}\\overline{\\hatϕ(n)}\\hat{f}(m+n)\\Bigr|^2. \\] However, the formula can break down for some other $f\\in\\mathcal{H}(b)$. In this article we extend the scope of the formula to all $f\\in H^2$ for which the right-hand side is finite, provided that either $ϕ\\in H^2$ or $ϕ$ is rational. If merely $ϕ\\in H^p$ for some $p\\in(0,2]$, then the formula still holds provided that, in addition, $\\sum_{m\\ge0}m^{2/p-1}|\\hat{f}(m)|^2<\\infty$. We also establish a limit-form of the formula that is valid for all non-extreme $b$ and all $f\\in\\mathcal{H}(b)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.30114",
  "author_total": 1,
  "authors": [
    {
      "name": "Thomas Ransford"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fjwjxn20nw0z2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.30114",
  "fulltext_url": null,
  "id": "fjwjxn20nw0z2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.30114",
  "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.CV"
    ],
    [
      "arxiv",
      "math.FA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2605.30114",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fjwjxn20nw0z2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2605.30114",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.30114"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the coefficient formula for de Branges-Rovnyak norms",
  "title_text": "On the coefficient formula for de Branges-Rovnyak norms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/fjwjxn20nw0z2",
  "venue": "arXiv",
  "versions": []
}