We consider $L^2$-approximation on weighted reproducing kernel Hilbert spaces of functions depending on infinitely many variables. We focus on unrestricted linear information, admitting evaluations of arbitrary continuous linear functionals. We distinguish between ANOVA and non-ANOVA spaces, where, by ANOVA spaces, we refer to function spaces whose norms are induced by an underlying ANOVA function decomposition. In ANOVA spaces, we provide an optimal algorithm to solve the approximation problem using linear information. We determine the upper and lower error bounds on the polynomial convergence rate of $n$-th minimal worst-case errors, which match if the weights decay regularly. For non-ANOVA spaces, we also establish upper and lower error bounds. Our analysis reveals that for weights with a regular and moderate decay behavior, the convergence rate of $n$-th minimal errors is strictly higher in ANOVA than in non-ANOVA spaces.

翻译：摘要：本文研究依赖于无穷多个变量的加权再生核希尔伯特空间上的 $L^2$ 逼近问题。我们关注无限制线性信息，允许对任意连续线性泛函进行评估。我们区分ANOVA空间与非ANOVA空间，其中ANOVA空间指范数由底层ANOVA函数分解所诱导的函数空间。在ANOVA空间中，我们提供了一种利用线性信息求解逼近问题的最优算法。我们给出了第 $n$ 最小最坏情况下误差的多项式收敛速度的上下界，当权重规则衰减时，两者是匹配的。对于非ANOVA空间，我们也建立了误差上下界。我们的分析表明，当权重具有规则且适度的衰减行为时，ANOVA空间中第 $n$ 最小误差的收敛速度严格高于非ANOVA空间。