奇数阶凸性的精细化Gauss--Lobatto界
Refined Gauss--Lobatto bounds for odd-order convexity
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中文总结 AI 辅助
本文针对奇数阶凸函数，证明积分位于Gauss--Legendre规则与其与Gauss--Lobatto规则中点之间，提出带尖锐常数1/4的认证估计器，并区分数值认证与高阶近似。
中文摘要 AI 辅助
对于高阶凸函数，经典极值性质将Gauss求积确定为积分的一个下界，而Gauss--Lobatto求积确定为积分的一个上界。我们在每个奇数阶上收紧了这一区间：若$f$在$[-1,1]$上是$(2n-1)$-凸的，则精确积分$I[f]:=\int_{-1}^{1}f(x)\\,dx$不仅位于$n$点Gauss--Legendre规则$G_n[f]$与$(n+1)$点Gauss--Lobatto规则$L_{n+1}[f]$之间，而且已经位于$G_n[f]$与它们的中间点之间。这产生了经过认证的估计器$Q_n[f]:=\frac{3}{4}G_n[f]+\frac{1}{4}L_{n+1}[f]$，满足$|I[f]-Q_n[f]|\leqslant\frac{1}{4}|L_{n+1}[f]-G_n[f]|$，且常数$\frac{1}{4}$是尖锐的。我们还记录了偶数阶的一个互补现象：对于$2n$-凸函数，Gauss--Radau端点规则仍然包围积分，但两个Radau规则的中间点不能产生单侧细化。这一障碍由奇异的、变号的Radau求积核解释。最后，我们将数值认证与普通的高阶近似区分开来：我们包含了来自样条、矩和统计模型的具体例子，并报告了截断幂和近极点Stieltjes核的实验，在这些情况下，当经典的基于光滑性的误差常数不可用或过于悲观时，形状证书仍然具有意义。
英文摘要
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.
发表机构
- Czech Academy of Sciences（捷克科学院）
- Institute of Mathematics and Computer Science Jagiellonian University（雅盖隆大学数学与计算机科学研究所）
- University of Bielsko–Biała（别尔斯科比亚拉大学）
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