Refined Gauss--Lobatto bounds for odd-order convexity
Abstract
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]:=∫−11f(x) dx lies not merely between the n-point Gauss--Legendre rule Gn[f] and the (n+1)-point Gauss--Lobatto rule Ln+1[f], but already between Gn[f] and their midpoint. This yields the certified estimator Qn[f]:=3/4Gn[f]+1/4Ln+1[f], with |I[f]−Qn[f]|leqslant1/4|Ln+1[f]−Gn[f]|, and the constant 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.
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BibTeX
@misc{kania2026refined,
title = {{Refined Gauss--Lobatto bounds for odd-order convexity}},
author = {Tomasz Kania and Szymon Wąsowicz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34918},
eprint = {2609.34918},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34918},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Refined Gauss--Lobatto bounds for odd-order convexity
AU - Tomasz Kania
AU - Szymon Wąsowicz
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34918
UR - https://arxiv.org/abs/2609.34918
AB - 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]:=∫_−1^1f(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]:=3/4G_n[f]+1/4L_n+1[f], with |I[f]−Q_n[f]|leqslant1/4|L_n+1[f]−G_n[f]|, and the constant 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.
ER -