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：我们考虑定义在依赖于无限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求值。我们区分ANOVA空间与非ANOVA空间，其中ANOVA空间指其范数由底层ANOVA函数分解诱导的函数空间。在ANOVA空间中，我们给出利用线性信息求解逼近问题的最优算法。我们确定了$n$阶最小最坏情况误差的多项式收敛速率的上下误差界，当权重呈现正则衰减时二者匹配。对于非ANOVA空间，我们也建立了上下误差界。我们的分析表明，对于具有正则且适度衰减行为的权重，ANOVA空间中$n$阶最小误差的收敛速率严格高于非ANOVA空间。