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：我们研究无穷多变量函数的积分与$L^2$逼近，设定如下：底层函数空间为单变量Hermite空间的可数无穷张量积，概率测度则为标准正态分布的相应乘积。该张量积空间中的函数最大定义域必然是序列空间$\mathbb{R}^\mathbb{N}$的一个真子集。我们在一般假设下建立了最小最坏情况误差的上下界；这些界对于具有有限或无限光滑性的常见Hermite函数空间之张量积是匹配的。在证明中，我们使用了嵌入结果，并通过多元分解方法实现了上界的构造性证明。