In this paper, we study bounds of expected $L_2-$discrepancy to give mean square error of uniform integration approximation for functions in Sobolev space $\mathcal{H}^{\mathbf{1}}(K)$, where $\mathcal{H}$ is a reproducing Hilbert space with kernel $K$. Better order $O(N^{-1-\frac{1}{d}})$ of approximation error is obtained, comparing with previously known rate $O(N^{-1})$ using crude Monte Carlo method. Secondly, we use expected $L_{p}-$discrepancy bound($p\ge 1$) of stratified samples to give several upper bounds of $p$-moment of integral approximation error in general Sobolev space $F_{d,q}^{*}$.
翻译：在本文中,我们研究了预期的2美元差异的界限,以给出Sobolev空间功能统一整合近似值的平均平方差值 $\ mathcal{H ⁇ mathbf{1}(K)$, 其中$\ mathcal{H} 是用核心美元复制Hilbert空间。 获得了更好的近似差值排序$O( N ⁇ -1-\frac{1 ⁇ d}, 与使用粗制蒙特卡洛法的已知率$O( ⁇ -1}) 相比。 其次,我们使用预计的分层样本的 $ ⁇ p}- $( p\ge 1 $) 来给Sobolev 空间整体近似差值以美元移动数个上限。 $F ⁇ d, q$。
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