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We study the numerical approximation of integrals over ${R}^s$ 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 ${R}^s$ 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.

On the Optimal Order of Integration in Hermite Spaces with Finite Smoothness