Sub-optimality of Gauss–Hermite quadrature and optimality of trapezoidal rule for functions with finite smoothness
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A sub-optimality of Gauss–Hermite quadrature and an 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 the matching lower and upper bounds, which turns 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 on 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.
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