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 prove that there is an optimal algorithm to solve the approximation problem using linear information. This way, we can determine the exact polynomial convergence rate of $n$-th minimal worst-case errors. For non-ANOVA spaces, we also establish upper and lower error bounds. Even though the bounds do not match in this case, they reveal that for weights with a moderate decay behavior, the convergence rate of $n$-th minimal errors is strictly higher in ANOVA than in non-ANOVA spaces.

翻译：我们考虑定义在加权再生核希尔伯特空间上的$L^2$逼近问题，其中函数依赖于无穷多个变量。重点研究无限制线性信息，即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：所谓ANOVA空间，是指其范数由底层ANOVA函数分解导出的函数空间。在ANOVA空间中，我们证明存在一种利用线性信息解决逼近问题的最优算法，从而能够精确确定$n$阶最小最坏情况误差的代数量收敛速率。对于非ANOVA空间，我们还建立了误差的上界和下界。尽管在此情形下上下界并不匹配，但它们揭示出：对于具有中等衰减行为的权重，ANOVA空间中$n$阶最小误差的收敛速率严格高于非ANOVA空间。