On the optimal order of integration in Hermite spaces with finite smoothness
Provides optimal-order integration rules for a class of smooth functions, advancing numerical analysis for Gaussian-weighted integrals.
The paper studies numerical integration over R^s with Gaussian measure for functions in Hermite spaces of finite smoothness, showing that higher-order digital nets achieve optimal convergence rates (up to log factors).
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.