AI 中文总结
本文利用随机最小二乘构造在函数子空间上精确的求积规则，通过新条件证明权重集中性与稳定性概率界，推动计算上构造性的广义Tchakaloff定理发展。
AI 中文摘要
我们考虑使用随机最小二乘来构造在函数子空间上精确的求积规则。利用我们称为相对可容许性和参考权重集中性的新条件，我们证明了此类过程产生的求积权重以预先指定的任意大概率集中在其渐近值附近，这反过来为非常一般的可能复值函数类提供了所得求积规则稳定性的有用有限样本概率界。我们的分析既显著推广了现有的随机最小二乘求积构造分析，又为这些规则的稳定性提供了新的界。这些结果专门化为正求积规则的存在性结果，当此类规则在理论上可预期时。由于我们的过程在形式上是算法性的，我们的分析是朝着计算上构造性的广义Tchakaloff定理的实质性进展。
英文摘要
We consider using randomized least squares to construct quadrature rules exact on a subspace of functions. Using new conditions that we call relative admissibility and reference weight concentration, we establish both that the quadrature weights from such a procedure concentrate close to their asymptotic values with prescribed and arbitrarily large probability, and this in turn provides useful finite-sample probabilistic bounds on the stability of the resulting quadrature rules for very general classes of possibly complex-valued functions. Our analysis both significantly generalizes the existing analysis of randomized least squares quadrature construction, and provides new bounds on stability for these rules. These results specialize to existence results for positive quadrature rules, when such rules can be theoretically expected. Because our procedures are formally algorithmic, our analysis is a substantive advance toward computationally constructive generalized Tchakaloff theorems.
