A sub-optimality of Gauss--Hermite quadrature and an optimality of the trapezoidal rule are proved in the weighted Sobolev spaces of square integrable functions of order $\alpha$, where the optimality is in the sense of worst-case error. For Gauss--Hermite quadrature, we obtain the matching lower and upper bounds, which turns out to be merely of the order $n^{-\alpha/2}$ with $n$ function evaluations, although the optimal rate for the best possible linear quadrature is known to be $n^{-\alpha}$. Our proof on the lower bound exploits the structure of the Gauss--Hermite nodes; the bound is independent of the quadrature weights, and changing the Gauss--Hermite weights cannot improve the rate $n^{-\alpha/2}$. In contrast, we show that a suitably truncated trapezoidal rule achieves the optimal rate up to a logarithmic factor.

翻译：Gaus-Hermite 二次曲线的亚优化度和 COOS- Hermite 二次曲线规则的优化性在按 $\\ alpha$ 排序的平方不可换函数的加权 Sobolev 空间 $\ alpha$ 中得到了证明。 对于高斯- Hermite 二次曲线来说,我们得到了匹配的下限和上界值,这被证明只是按 $ ⁇ -\ alpha/2 美元 的排序, 并附有 $n 美元 函数评价, 尽管已知最佳的线性二次曲线矩形的最佳比率是 $ ⁇ -\ alpha} 。 我们关于下界点的证明利用了高斯- Hermite 节点的结构; 约束独立于二次曲线的重量, 并且改变高斯- 赫米特的重量不能改善 $n ⁇ - alpha/2 美元 的速率 。 相反, 我们表明, 一种适当的三角三角三角矩定 规则达到对数系数的最佳比率。