We investigate the approximation of weighted integrals over $\mathbb{R}^d$ for integrands from weighted Sobolev spaces of mixed smoothness. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to $n$ integration nodes for functions from these spaces. In the one-dimensional case $(d=1)$, we obtain the right convergence rate of optimal quadratures. For $d \ge 2$, the upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic crosses in the function domain $\mathbb{R}^d$.

翻译：本文研究了来自混合光滑性加权Sobolev空间的被积函数在$\mathbb{R}^d$上的加权积分逼近问题。我们证明了这些空间中函数在$n$个积分节点下最优求积公式收敛速度的上界和下界。在一维情形$(d=1)$下，我们得到了最优求积公式的准确收敛速度。对于$d \ge 2$，上界通过稀疏网格求积公式实现，其积分节点位于函数定义域$\mathbb{R}^d$上的阶梯双曲交叉区域中。