中文摘要：

本文讨论了在Wiener空间下的最优求积公式在r-重积分Wiener空间下的平均误差,得到了相应量的值或强渐近阶,结果证明该求积公式在平均误差情形下具有饱和性。本文的结果说明了此求积公式虽对Wiener空间是最优的,但对1-重积分Wiener空间仅仅是阶最优的,而当r≥2时,此求积公式在r-重积分Wiener空间下没有任何最优性。因此,对于计算具有不同光滑性的函数的积分而言,此积分公式不是普适算法。

英文摘要：

In this paper, the average errors of optimal quadrature formula in the Wiener space are discussed on the r-fold integrated Wiener space, lts values or strongly asymptotic order is obtained. From the results it is shown that these formulas have saturation property on the average error case. Our results show that T,, are only order optimal for 1-fold integrated Wiener Space, and is not optimal for r-fold integrated Wiener Space when r≥2. Hence, for the computation of integral of functions with different smooth- ness, Tn are not universal operator.