Suboptimality of Gauss–Hermite quadrature and optimality of the trapezoidal rule for functions with finite smoothness

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 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.