论文标题
高斯的亚次临时 - 甲虫的正交和梯形规则的最佳性，具有有限平滑度的功能
Sub-optimality of Gauss--Hermite quadrature and optimality of the trapezoidal rule for functions with finite smoothness
论文作者
论文摘要
高斯 - 热线正交和梯形规则的最佳性的子次要性在加权的sobolev空间中证明了$α$的正方形集成函数，其中最优性在最坏情况下是最优点。对于高斯 - 热线正交，我们获得了匹配的下限和上限，事实证明，这些边界仅适用于$ n^{ - α/2} $带有$ n $函数评估的顺序，尽管已知最佳线性正交的最佳速率是$ n^{ - α} $。我们对下边界的证明利用了高斯 - 热点节点的结构；界限独立于正交权重，并且更改高斯 - 荷米矿的权重无法提高速率$ n^{ - α/2} $。相比之下，我们表明，适当截断的梯形规则可达到对数因子的最佳速率。
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 $α$, 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 $n^{-α/2}$ with $n$ function evaluations, although the optimal rate for the best possible linear quadrature is known to be $n^{-α}$. 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 $n^{-α/2}$. In contrast, we show that a suitably truncated trapezoidal rule achieves the optimal rate up to a logarithmic factor.
