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The sub-optimality of Gauss--Hermite quadrature and the optimality of thetrapezoidal rule are proved in the weighted Sobolev spaces of square integrablefunctions of order $\alpha$, where the optimality is in the sense of worst-caseerror. For Gauss--Hermite quadrature, we obtain matching lower and upperbounds, which turn out to be merely of the order $n^{-\alpha/2}$ with $n$function evaluations, although the optimal rate for the best possible linearquadrature is known to be $n^{-\alpha}$. Our proof of the lower bound exploitsthe structure of the Gauss--Hermite nodes; the bound is independent of thequadrature weights, and changing the Gauss--Hermite weights cannot improve therate $n^{-\alpha/2}$. In contrast, we show that a suitably truncatedtrapezoidal rule achieves the optimal rate up to a logarithmic factor.

Sub-optimality of Gauss--Hermite quadrature and optimality of the trapezoidal rule for functions with finite smoothness