On the optimal order of integration in Hermite spaces with finite smoothness

Abstract

We study the numerical approximation of integrals over ${\mathbb{R}}^s$ with respect to the standard Gaussian measure for integrands which lie in certain Hermite spaces of functions. The decay rate of the associated sequence is specified by a single integer parameter which determines the smoothness classes and the inner product can be expressed via $L_2$ norms of the derivatives of the function. We map higher order digital nets from the unit cube to a suitable subcube of ${\mathbb{R}}^s$ via a linear transformation and show that such rules achieve, apart from powers of $\log N$, the optimal rate of convergence of the integration error.

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