Optimal Integration of Lipschitz Functions with a Gaussian Weight

Abstract

We study optimal integration over the infinite interval (-∞, +∞) for a Gaussian weight with variance σ. We consider functions satisfying a Lipschitz condition with constant L. We characterize the optimal information by a system of nonlinear equations, and show that (asymptotically) the solution is defined by the quantiles of the Gaussian weight with double variance 2σ. We provide an optimal algorithm and prove that for n optimal sample points the worst case error of the optimal algorithm is asymptotically equal to √πσ/2L/n. Finally, we show that the worst case error for the set of sampling points defined by the zeros of Hermite polynomials is quadratically worse than for optimal sample points. © 1998 Academic Press.