We prove that a variant of the classical Sobolev space of first-order dominating mixed smoothness is equivalent (under a certain condition) to the unanchored ANOVA space on $\mathbb{R}^d$, for $d \geq 1$. Both spaces are Hilbert spaces involving weight functions, which determine the behaviour as different variables tend to $\pm \infty$, and weight parameters, which represent the influence of different subsets of variables. The unanchored ANOVA space on $\mathbb{R}^d$ was initially introduced by Nichols & Kuo in 2014 to analyse the error of quasi-Monte Carlo (QMC) approximations for integrals on unbounded domains; whereas the classical Sobolev space of dominating mixed smoothness was used as the setting in a series of papers by Griebel, Kuo & Sloan on the smoothing effect of integration, in an effort to develop a rigorous theory on why QMC methods work so well for certain non-smooth integrands with kinks or jumps coming from option pricing problems. In this same setting, Griewank, Kuo, Le\"ovey & Sloan in 2018 subsequently extended these ideas by developing a practical smoothing by preintegration technique to approximate integrals of such functions with kinks or jumps. We first prove the equivalence in one dimension (itself a non-trivial task), before following a similar, but more complicated, strategy to prove the equivalence for general dimensions. As a consequence of this equivalence, we analyse applying QMC combined with a preintegration step to approximate the fair price of an Asian option, and prove that the error of such an approximation using $N$ points converges at a rate close to $1/N$.

翻译：我们证明,古典的Sobolev 空间的变体,即一流的混合平滑度,相当于(在一定条件下)以$$\mathbb{R ⁇ d$美元为单位的ANOVA无序空间。两个空间都是涉及重量函数的Hilbert空间,确定行为,因为不同的变量往往为$pm\infty$,重量参数代表不同变量组别的影响。Nichols & Kuo在2014年首次推出(在一定条件下)ANOVA空间,等于(在一定条件下)以美元为单位的ANOVA空间,以分析准Monte Car(QMC)在未受约束域的 ANOVA空间的偏差;而Griebel、Ku & Sloan在一系列论文前的设定行为,以美元为单位的平滑滑度为单位,试图形成一个精确的QMC方法的精度理论,但以美元为单位的直径直径直, 以直径直的直径直的直径直直径直径直径直径直径直直直向方向直径直径直径直径直径直径直向方向的策略, 将类似的Lebil 将一个方向直径直直向一个方向直向一个方向直向一个方向向一个方向向一个方向直向一个方向, 。