Mathematical Sciences: Topics in Ergodic Theory and Analysis

数学科学：遍历理论与分析主题

• 批准号: 8910947
• 负责人: Alexandra Bellow
• 金额: $ 8.55万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-15 至 1991-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8910947&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topics; Ergodic; Theory

Professors Bellow and Jones will continue their ongoing investigation of almost-everywhere convergence questions arising in connection with the pointwise ergodic theorem along subsequences and along other sequences of operators. A.e. convergence in mean for certain arithmetic sequences will be an especial focus of interest. Calderon-Zygmund type decomposition techniques will be used to study such problems. The principal investigators will also pursue problems at the other extreme involving the strong sweeping out property, where a.e. convergence in the sup norm fails very badly. This project is mathematical research in ergodic theory. This theory is concerned with what happens on average over the long run when a suitable transformation of a suitable underlying space is iterated many times. (Typically, the underlying space might be just a line segment, and the transformation might cut the line segment up into pieces, then stretch or shrink and rearrange them.) One point of view is to study the behavior of functions on the space when their arguments are repeatedly subjected to the transformation. It frequently happens that the averages of the iterates of the functions converge in a meaningful fashion. A recent trend in the subject is to look at averages over subsequences, for instance the squares or the primes, instead of over all integer indices. Professors Bellow and Jones will pursue ergodic theory over subsequences in this sense.

贝娄和琼斯教授将继续他们正在进行的关于沿子序列和其他算子序列的逐点遍历定理所产生的几乎处处收敛问题的研究。A.E.某些算术序列的平均收敛将是一个特别感兴趣的焦点。Calderon-Zygmund型分解技术将被用来研究这些问题。主要调查人员还将调查另一个极端的问题，涉及强大的横扫财产，其中A.E.在超范数下的收敛非常失败。这个项目是遍历理论中的数学研究。这一理论关注的是，当一个合适的底层空间的适当转换被多次迭代时，长期平均会发生什么。(通常，底层空间可能只是一个线段，转换可能会将线段分割成多个片段，然后拉伸或收缩并重新排列它们。)一种观点是研究函数在空间上的行为，当它们的自变量反复经历变换时。经常会发生函数迭代的平均值以有意义的方式收敛的情况。这一主题最近的一个趋势是查看子序列的平均值，例如平方或质数，而不是所有整数指数。在这个意义上，贝娄和琼斯教授将继续研究子序列的遍历理论。