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A Computationally-Efficient Closed-Form $C^{s,1}$-Extension Formula — arXiv:2610.12239

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.12239
License (as archived): CC0-1.0
Archived: 2026-10-09T06:29:49.509Z
SHA-256 of the archived text: b4ead09fc048584082137c8f3f0bebee169c194163e2186ce90c82668fb01c8f
Stable URL of this page: https://metraipolis.com/documento/dc_c4ecc4eaf649

How to cite

A Computationally-Efficient Closed-Form $C^{s,1}$-Extension Formula — arXiv:2610.12239 — arXiv (en), https://arxiv.org/abs/2610.12239 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_c4ecc4eaf649 (Metraipolis archive dc_c4ecc4eaf649, sha256 b4ead09fc048584082137c8f3f0bebee169c194163e2186ce90c82668fb01c8f)

BibTeX

@misc{dcc4ecc4eaf649,
  title = {A Computationally-Efficient Closed-Form $C^s,1$-Extension Formula — arXiv:2610.12239},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.12239},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_c4ecc4eaf649; sha256 b4ead09fc048584082137c8f3f0bebee169c194163e2186ce90c82668fb01c8f},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.12239. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.LO, nuovi lavori del giorno).
Categorie arXiv: Functional Analysis (math.FA) ; Data Structures and Algorithms (cs.DS); Classical Analysis and ODEs (math.CA); Logic (math.LO).
Pagina del lavoro: https://arxiv.org/abs/2610.12239 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-09T05:54:21.511Z dalla pagina https://arxiv.org/list/math.LO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).