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 be either 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 [\textit{Numer. Math.}, 75 (1997), pp.~379--395] and provide exceptionally sharp lower bounds apart from a polynomial factor, in particular show that the worst-case error for the trapezoidal rule by Sugihara is not improvable 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, Gauss--Hermite quadrature is proven sub-optimal 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 [\textit{SIAM Rev.}, 64 (2022), pp.~132--150] for the upper error bounds in terms of the decay conditions.

翻译：本文研究了解析函数在实直线上的数值积分问题。我们重点关注误差界的锐利程度。首先推导了两个关于最坏情况积分误差的通用下界估计，随后将其应用于建立多种求积规则的下界。这些下界或是全新的，或是对现有结果的改进，使得各公式的下界与上界高度吻合。具体而言，针对适当截断的梯形公式，我们改进了Sugihara [\textit{Numer. Math.}, 75 (1997), pp.~379--395] 获得的最坏情况误差通用下界，并给出了除多项式因子外极其锐利的下界，特别证明了Sugihara梯形公式的最坏情况误差无法在超过多项式因子的范围内改进。此外，我们的研究揭示了梯形公式的误差衰减性与Sugihara针对通用数值积分公式所获下界之间的差异，从而引出一个新的开放问题。同时，在考虑的被积函数衰减条件下，证明了Gauss--Hermite求积公式的非最优性，这一结论无法仅从上界论证推导得出。进一步地，为建立适当尺度化Gauss--Legendre和Clenshaw--Curtis求积公式的近似最优性，我们推广了Trefethen [\textit{SIAM Rev.}, 64 (2022), pp.~132--150] 最近关于基于衰减条件导出误差上界的结果。