SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2370-2392, October 2024.
Abstract. Numerical integration over the real line for analytic functions is studied. Our main focus is on the sharpness of the error bounds. We first derive two general lower estimates for the worst-case integration error, and then apply these to establish lower bounds for various quadrature rules. These bounds turn out to either be novel or improve upon existing results, leading to lower bounds that closely match upper bounds for various formulas. Specifically, for the suitably truncated trapezoidal rule, we improve upon general lower bounds on the worst-case error obtained by Sugihara [Numer. Math., 75 (1997), pp. 379–395] and provide exceptionally sharp lower bounds apart from a polynomial factor, and in particular we show that the worst-case error for the trapezoidal rule by Sugihara is not improvable by more than a polynomial factor. Additionally, our research reveals a discrepancy between the error decay of the trapezoidal rule and Sugihara’s lower bound for general numerical integration rules, introducing a new open problem. Moreover, the Gauss–Hermite quadrature is proven suboptimal under the decay conditions on integrands we consider, a result not deducible from upper-bound arguments alone. Furthermore, to establish the near-optimality of the suitably scaled Gauss–Legendre and Clenshaw–Curtis quadratures, we generalize a recent result of Trefethen [SIAM Rev., 64 (2022), pp. 132–150] for the upper error bounds in terms of the decay conditions.

中文翻译：

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误差范围有多尖锐？–解析函数的正交最坏情况误差下限–


SIAM 数值分析杂志，第 62 卷，第 5 期，第 2370-2392 页，2024 年 10 月。

抽象。研究了解析函数在实线上的数值积分。我们主要关注误差边界的锐度。我们首先为最坏情况的积分误差推导出两个一般的下限估计值，然后应用它们来建立各种正交规则的下限。这些边界要么是新颖的，要么是对现有结果的改进，导致下限与各种公式的上限紧密匹配。具体来说，对于适当截断的梯形规则，我们改进了 Sugihara 获得的最坏情况误差的一般下限 [Numer. Math.， 75 （1997）， pp. 379–395]，并提供了除多项式因子之外的异常陡峭的下限，特别是我们表明 Sugihara 的梯形规则的最坏情况误差不能由多项式因子改善。此外，我们的研究揭示了梯形规则的误差衰减与一般数值积分规则的 Sugihara 下限之间存在差异，引入了一个新的开放问题。此外，在我们考虑的被积函数的衰减条件下，Gauss-Hermite 求积被证明是次优的，这一结果不能仅从上限参数中推导出来。此外，为了建立适当缩放的高斯-勒让德和克伦肖-柯蒂斯求积的近最优性，我们推广了 Trefethen 的最新结果 [SIAM Rev.， 64 （2022）， pp. 132–150] 关于衰减条件的误差上限。