We study integration and $L^2$-approximation of functions of infinitely many variables in the following setting: The underlying function space is the countably infinite tensor product of univariate Hermite spaces and the probability measure is the corresponding product of the standard normal distribution. The maximal domain of the functions from this tensor product space is necessarily a proper subset of the sequence space $\mathbb{R}^\mathbb{N}$. We establish upper and lower bounds for the minimal worst case errors under general assumptions; these bounds do match for tensor products of well-studied Hermite spaces of functions with finite or with infinite smoothness. In the proofs we employ embedding results, and the upper bounds are attained constructively with the help of multivariate decomposition methods.

翻译：我们在以下环境中研究具有无限多个变量的函数的积分和 $L^2$-近似:底层函数空间是单变量 Hermite 空间的可数次张量积，概率测度是相应的标准正态分布的乘积。由于函数的最大定义域来自这种张量积空间且为序列空间 $\mathbb{R}^\mathbb{N}$ 的一个子集，因此必然是恰当的。在一般前提下，我们建立了最小最差误差的上下界；这些边界对于具有有限或无限光滑性的 Hermite 函数空间的张量积来说是匹配的。在证明中，我们采用嵌入结果，上界是通过多元分解方法进行建模的。