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Refined Gauss--Lobatto bounds for odd-order convexity — arXiv:2609.34918

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.34918
License (as archived): CC0-1.0
Archived: 2026-10-07T23:25:18.436Z
SHA-256 of the archived text: 052c2352de5074335f2e10a58422c43dff6fa0b71faa6cf5c2764590d37d0de0
Stable URL of this page: https://metraipolis.com/documento/dc_786fdf4e4be2

How to cite

Refined Gauss--Lobatto bounds for odd-order convexity — arXiv:2609.34918 — arXiv (en), https://arxiv.org/abs/2609.34918 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_786fdf4e4be2 (Metraipolis archive dc_786fdf4e4be2, sha256 052c2352de5074335f2e10a58422c43dff6fa0b71faa6cf5c2764590d37d0de0)

BibTeX

@misc{dc786fdf4e4be2,
  title = {Refined Gauss--Lobatto bounds for odd-order convexity — arXiv:2609.34918},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.34918},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_786fdf4e4be2; sha256 052c2352de5074335f2e10a58422c43dff6fa0b71faa6cf5c2764590d37d0de0},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.34918. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

For functions that are convex of higher order, classical extremalities identify Gaussian quadrature as a lower bound and Gauss--Lobatto quadrature as an upper bound for the integral. We sharpen this bracket in every odd order: if $f$ is $(2n-1)$-convex on $[-1,1]$, then the exact integral $I[f]:=\int_{-1}^{1}f(x)\,dx$ lies not merely between the $n$-point Gauss--Legendre rule $G_n[f]$ and the $(n+1)$-point Gauss--Lobatto rule $L_{n+1}[f]$, but already between $G_n[f]$ and their midpoint. This yields the certified estimator $Q_n[f]:=\frac{3}{4}G_n[f]+\frac{1}{4}L_{n+1}[f]$, with $|I[f]-Q_n[f]|\leqslant\frac{1}{4}|L_{n+1}[f]-G_n[f]|$, and the constant $\frac{1}{4}$ is sharp. We also record a complementary phenomenon in even order: for $2n$-convex functions the Gauss--Radau endpoint rules still bracket the integral, but the midpoint of the two Radau rules cannot yield a one-sided refinement. The obstruction is explained by an odd, sign-changing Radau quadrature kernel. Finally, we distinguish numerical certification from ordinary high-order approximation: we include concrete examples from spline, moment and statistical models and report experiments for truncated powers and near-pole Stieltjes kernels, where the shape certificate remains meaningful when classical smoothness-based error constants are unavailable or severely pessimistic.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-29 in math.NA).
Categorie arXiv: Numerical Analysis (math.NA) ; Classical Analysis and ODEs (math.CA).
Pagina del lavoro: https://arxiv.org/abs/2609.34918 — 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 2026-10-07T23:04:09.508Z dalla pagina https://arxiv.org/catchup/math.NA/2026-09-29?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).