{
  "abstract": "We identify explicit closed-form and variational formulae for interpolating the exact values and derivatives through order $s$ of a $C^{s,1}$ function $f:\\mathbb{R}^d\\to\\mathbb{R}$ at $N$ distinct points in $[0,1]^d$. The reconstruction satisfies bounds on its global $C^{s,1}$ seminorm and its Lipschitz constant on $[0,1]^d$ that are independent of the sample size $N$, the latter being the sharp Whitney condition. For $s\\ge2$, the reconstruction is real analytic away from the data points and definable in the o-minimal structure $\\mathbb{R}_{\\exp}$. Our nonlinear extension formula coincides with the formula of McShane (1934) for $s=0$ and with the formulae of Le Gruyer and Phan (2015) and Azagra, Le Gruyer, and Mudarra (2018) for $s=1$, and is new for $s\\ge2$. For fixed $d$ and $s\\ge2$, our closed-form formula is computable away from the data points by a circuit using only elementary unary and binary real operations, with $\\mathcal{O}(N)$ gates and $\\mathcal{O}(\\log N)$ depth. This yields $\\mathcal{O}(N)$ storage and $\\mathcal{O}(\\log N)$ parallel evaluation time. From the supplied coefficients and weights, the circuit can be compiled with $\\mathcal{O}(N)$ one-time work in the exact-real word-RAM model. Thus, in this supplied-data setting, our nonlinear construction improves by a logarithmic factor on the $\\mathcal{O}(N\\log N)$ initialization bound of Fefferman and Klartag (2009), while retaining linear storage and attaining logarithmic query time through parallel evaluation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12239",
  "author_total": 2,
  "authors": [
    {
      "name": "Anastasis Kratsios"
    },
    {
      "name": "Philipp Zimmermann"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/91gzwd9q34dak.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12239",
  "fulltext_url": null,
  "id": "91gzwd9q34dak",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12239",
  "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",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
  ],
  "openalex": "W7221125022",
  "pdf_url": "https://arxiv.org/pdf/2610.12239",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/91gzwd9q34dak.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.12239",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12239"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "A Computationally-Efficient Closed-Form $C^{s,1}$-Extension Formula",
  "title_text": "A Computationally-Efficient Closed-Form C^s,1-Extension Formula",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/91gzwd9q34dak",
  "venue": "arXiv",
  "versions": []
}