{
  "abstract": "The variational theory of splines requires a native space: a Hilbert space over which the regularization energy $\\|\\mathrm{L} f\\|_{L_2}^2$ is well defined and the minimum-energy interpolation problem admits a spline solution. In this paper, we characterize the operators $\\mathrm{L}$ that enable the construction of such spaces. Our framework is based on the completion of a space $\\mathcal{S}$ of test functions in a controlled Gelfand-triple configuration $\\mathcal{S}\\subset\\mathcal{H}\\subset\\mathcal{S}'$, which we call a Hilbert-Gelfand space. This departs from the classical reproducing-kernel techniques. Our construction is substantially more general, as it supports arbitrary linear operators and singular behaviors, and it recovers the reproducing-kernel setting as a particular case. It is also complete and canonical: the admissibility conditions on $\\mathrm{L}$ are necessary and sufficient, and $\\mathrm{L}$ determines its native space as the unique Hilbert-Gelfand space on which it extends to a bounded surjection onto $L_2$. We first show that Hilbert-Gelfand spaces are in one-to-one correspondence with compatible Riesz pairs $(\\mathrm{A},\\mathrm{A}^{-1})$ of positive-definite operators $\\mathcal{S}\\to\\mathcal{S}'$. We then extend this correspondence to conditional Riesz pairs, where $\\mathrm{A}$ is only conditionally positive-definite, which is the scenario that is relevant to splines. The theory identifies the \\emph{attainable null space} of the energy as the unique admissible conditioning, realizes the abstract completions as concrete spaces of distributions, and provides a verifiable admissibility condition in terms of a dual energy functional. The framework is illustrated with the fractional Laplacian, which covers polyharmonic and thin-plate splines, and with ordinary differential operators, which cover exponential splines.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14906",
  "author_total": 1,
  "authors": [
    {
      "name": "Michael Unser"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7gzstfm80jf65.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14906",
  "fulltext_url": null,
  "id": "7gzstfm80jf65",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14906",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7221800526",
  "pdf_url": "https://arxiv.org/pdf/2610.14906",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7gzstfm80jf65.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14906",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14906"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Splines and Operators: Part I - Admissibility and Native Spaces",
  "title_text": "Splines and Operators: Part I - Admissibility and Native Spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/7gzstfm80jf65",
  "venue": "arXiv",
  "versions": []
}