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：我们研究以下设定中无限多变量函数的积分与$L^2$逼近问题：底层函数空间为可数无限个单变量Hermite空间的张量积，概率测度为标准正态分布的相应乘积。该张量积空间中函数的最大定义域必是序列空间$\mathbb{R}^\mathbb{N}$的真子集。我们建立了在一般假设下最小最坏情况误差的上下界；对于具有有限或无限光滑性的常见Hermite函数空间张量积，这些界恰好匹配。在证明中，我们使用了嵌入结果，并通过多元分解方法构造性地达到了上界。