The sub-optimality of Gauss--Hermite quadrature and the 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 matching lower and upper bounds, which turn 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 of 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:高斯- 赫米特二次曲线的亚最佳度和角化规则的最佳性在按正方正方正方数正方数的正方正方正方正方正方方形面积 $\ alpha$ 的加权 Sobolev 空间中得到了证明, 最佳性在最坏的误差意义上是最佳的。 对于高斯- 赫米特二次曲线, 我们得到的比对下方和上方界限, 其结果仅仅是按 $ ⁇ -\ alpha/2 美元 的排序, 并有 $n ⁇ - alpha/2 的函数评价, 尽管已知最佳线性线性二次曲线的最佳率是 $ {\\\ alpha} 。 我们关于下界的证明是利用了高斯- 赫米特节点的结构; 约束独立于二次曲线重量, 并且改变高斯- 赫米特重量不能改善 $n ⁇ - ALpha/2 美元 的速率 。 相反, 我们显示, 一种适当的三角三角定式的梯度定最佳率达到对数系数。