a b s t r a c tWe study integration in a class of Hilbert spaces of analytic functions defined on the R s . The functions are characterized by the property that their Hermite coefficients decay exponentially fast. We use Gauss-Hermite integration rules and show that the errors of our algorithms decay exponentially fast. Furthermore, we study tractability in terms of s and log ε −1 and give necessary and sufficient conditions under which we achieve exponential convergence with EC-weak, EC-polynomial, and EC-strong polynomial tractability.

We study multivariate approximation defined over tensor product Hilbert spaces. The domain space is a weighted tensor product Hilbert space with exponential weights which depend on two sequences a = {a j } j∈N and b = {b j } j∈N of positive numbers, and on a bounded sequence of positive integers m = {m j } j∈N . The sequence a is non-decreasing and the sequence b is bounded from below by a positive number. We find necessary and sufficient conditions on a, b and m to achieve the standard and new notions of tractability in the worst case setting.

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.

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