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 ${\mathbb{R}}^d$.

中文翻译：

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具有混合平滑度的函数的数值加权积分

我们研究加权积分的近似值${\mathbb{R}}^d$对于来自混合平滑度的加权 Sobolev 空间的被积函数。我们证明了关于来自这些空间的函数的n 个积分节点的最佳正交收敛速度的上限和下限。在一维情况下$(d=\mathrm{1个})$，我们获得了最佳正交的正确收敛速度。为了$d\ge \mathrm{2个}$，上界由稀疏网格正交执行，积分节点位于函数域中的步长双曲交叉上${\mathbb{R}}^d$.