The paper deals with linear problems defined on $\mathit{\gamma }$-weighted Hilbert spaces of functions with infinitely many variables. The spaces are endowed with zero-mean Gaussian measures which allows to define and study $\epsilon$-truncation and $\epsilon$-superposition dimensions in the average case and probabilistic settings. Roughly speaking, these $\epsilon$-dimensions quantify the smallest number $k=k\left(\epsilon \right)$ of variables that allow to approximate the $\infty$-variate functions by special ones that depend on at most $k$-variables with the average error bounded by $\epsilon$. In the probabilistic setting, given $\delta \in (0,1)$, we want the error $\le \epsilon$ with probability $\ge 1-\delta$. We show that the $\epsilon$-dimensions are surprisingly small which, for anchored spaces, leads to very efficient algorithms, including the Multivariate Decomposition Methods.

本文讨论了定义在 $\mathit{\gamma }$具有无限多个变量的函数的加权希尔伯特空间。这些空间具有零均值高斯度量，可以用来定义和研究$\epsilon$-截断和 $\epsilon$-在平均情况和概率设置中的叠加维度。粗略地说，这些$\epsilon$尺寸量化最小数量 $ķ=ķ（\epsilon ）$ 变量的近似值 $\infty$-最多依赖的特殊函数的变量函数 $ķ$-平均误差范围为-的变量 $\epsilon$。在概率环境中，给定$\delta \in （0，\mathrm{1个}）$，我们想要的错误 $\le \epsilon$ 很有可能 $\ge \mathrm{1个}-\delta$。我们表明$\epsilon$尺寸小得令人惊讶，对于锚定空间，这导致了非常有效的算法，包括多元分解方法。