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We prove that a variant of the classical Sobolev space of first-orderdominating mixed smoothness is equivalent (under a certain condition) to theunanchored ANOVA space on $\mathbb{R}^d$, for $d \geq 1$. Both spaces areHilbert spaces involving weight functions, which determine the behaviour asdifferent variables tend to $\pm \infty$, and weight parameters, whichrepresent the influence of different subsets of variables. The unanchored ANOVAspace on $\mathbb{R}^d$ was initially introduced by Nichols & Kuo in 2014 toanalyse the error of quasi-Monte Carlo (QMC) approximations for integrals onunbounded domains; whereas the classical Sobolev space of dominating mixedsmoothness 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 arigorous theory on why QMC methods work so well for certain non-smoothintegrands with kinks or jumps coming from option pricing problems. In thissame setting, Griewank, Kuo, Le\"ovey & Sloan in 2018 subsequently extendedthese ideas by developing a practical smoothing by preintegration technique toapproximate 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 theequivalence for general dimensions. As a consequence of this equivalence, weanalyse applying QMC combined with a preintegration step to approximate thefair price of an Asian option, and prove that the error of such anapproximation using $N$ points converges at a rate close to $1/N$.

Equivalence between Sobolev spaces of first-order dominating mixed smoothness and unanchored ANOVA spaces on $\mathbb{R}^d$