切比雪夫型求积公式的简单通用界
Simple Universal Bounds for Chebyshev-Type Quadratures
AI总结：
针对区间上支撑的概率测度，利用矩估计给出切比雪夫型求积公式所需最少节点数的上下界，并应用于球面与圆柱上的点集构造。
摘要：
对于概率测度σ的切比雪夫型求积公式是n个点上均匀分布且与σ具有相同前k阶矩的分布。我们给出了在σ支撑于区间上时，达到给定阶数k所需最小n的上界。与以往此类结果相比，我们的界仅使用σ的简单性质，且广泛适用。我们还得到了所需节点数的下界，仅使用σ的矩估计。给出了说明界紧性的例子。作为结果的推论，我们得到了关于高斯求积的一个看似新的结果。此外，我们提出了另一种通过随机选择节点来界定切比雪夫型求积所需最少节点数的方法，并提出了分析其性能的挑战。对于立方体上的均匀测度，证明了该方向的一个初步结果。最后，我们将我们的界应用于球面和圆柱上构成局部近似切比雪夫型求积的点集构造。这些结果最近在理解泊松过程如何近似某些连续分布的背景下是必需的。论文以一系列开放问题结束。
英文摘要：
A Chebyshev-type quadrature for a probability measure sigma is a distribution which is uniform on n points and has the same first k moments as sigma. We give an upper bound for the minimal n required to achieve a given degree k, for sigma supported on an interval. In contrast to previous results of this type, our bound uses only simple properties of sigma and is applicable in wide generality. We also obtain a lower bound for the required number of nodes which only uses estimates on the moments of sigma. Examples illustrating the sharpness of our bounds are given. As a corollary of our results, we obtain an apparently new result on the Gaussian quadrature. In addition, we suggest another approach to bounding the minimal number of nodes required in a Chebyshev-type quadrature, utilizing a random choice of the nodes, and propose the challenge of analyzing its performance. A preliminary result in this direction is proved for the uniform measure on the cube. Finally, we apply our bounds to the construction of point sets on the sphere and cylinder which form local approximate Chebyshev-type quadratures. These results were needed recently in the context of understanding how well can a Poisson process approximate certain continuous distributions. The paper concludes with a list of open questions.
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