Suboptimality of Gauss–Hermite Quadrature and Optimality of the Trapezoidal Rule for Functions with Finite Smoothness

作者：Yoshihito Kazashi, Yuya Suzuki, Takashi Goda · 发表于：SIAM Journal on Numerical Analysis · 年份：2023 · DOI：10.1137/22m1480276 · 被引用次数：13 · 研究领域：Mathematical Approximation and Integration、Mathematical functions and polynomials、Advanced Numerical Methods in Computational Mathematics

Abstract. The suboptimality of Gauss–Hermite quadrature and the optimality of the trapezoidal rule are proved in the weighted Sobolev spaces of square integrable functions of order [Formula: see text], 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 [Formula: see text] with [Formula: see text] function evaluations, although the optimal rate for the best possible linear quadrature is known to be [Formula: see text]. 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 [Formula: see text]. In contrast, we show that a suitably truncated trapezoidal rule achieves the optimal rate up to a logarithmic factor.