Some problems in time-frequency analysis

这个项目将探索调制不变时频分析领域中的几个相互关联的问题。第一组问题是关于保测量系统的返回时间现象。这里，人们感兴趣的是在动力系统中沿着流的加权平均的收敛，特别是当权重由第二动力系统中的流产生时。两个特别令人感兴趣的问题是确定收敛所需的Lp正则性的量，特别是当平均值被奇异积分取代时，以及根据变分范数估计确定这种收敛发生得多快。这些问题应该通过证明Carleson-Hunt定理的适当推广来解决。第二组问题涉及双线性希尔伯特变换。这里感兴趣的一个问题是找到在决定变换奇点集的参数中一致的新估计。第二个问题是确定截断变换和相关极大算子的收敛速度。尽管乍一看，这两组问题似乎没有关联，但预计它们的解决在很大程度上将由一套共同的工具来指导。这项拟议的研究属于调和分析这一宽泛的范畴，这是一门研究将许多类型的数据分解成不同频率的基波组合的学科。这些分解在物理、化学、生物、金融等领域得到了广泛的应用。这里特别相关的一个应用是数据压缩，在这种应用中，人们寻求在使用最少的存储空间的同时准确地表示信号，例如电影或音轨。收敛速度估计，如上一段所述，原则上可以用来表征需要多少空间才能给出一个信号的“良好”表示。该项目还与调和分析以外的数学领域进行了重要的接触，包括遍历理论和加法组合学，预计将激发与路易斯安那州立大学数学系的师生以及全国各地的合作者进行许多积极的讨论。

This project will explore several interconnected questions in the area of modulation-invariant time-frequency analysis. The first group of questions is concerned with the return times phenomenon for measure-preserving systems. Here, one is interested in the convergence of weighted averages along flows in dynamical systems, specifically when the weights are generated by a flow in a second dynamical system. Two problems of particular interest are to determine the amount of Lp regularity required for convergence, especially when the averages are replaced by singular integrals, and to determine, in terms of variation-norm estimates, how quickly this convergence occurs. These problems should be addressed by proving suitable extensions of the Carleson-Hunt theorem. The second group of questions concerns the bilinear Hilbert transform. One problem of interest here is to find new estimates which are uniform in the parameter which determines the singularity set for the transform. A second problem is to determine the rate of convergence for truncated transforms and the associated maximal operator. Although, at first glance, these two groups of questions may not seem to be related, their resolution is expected to be largely guided by a common set of tools. The proposed research falls under the broad umbrella of harmonic analysis, a discipline which studies the decomposition of many types of data into combinations of basic waves of varying frequencies. These decompositions have been broadly applied in areas such as physics, chemistry, biology, and finance. An application which is especially relevant here is data compression, where one seeks to represent a signal, such as a movie or audio track, accurately while using a minimum amount of storage space. Convergence rate estimates, as in the previous paragraph, can in principle be used to characterize how much space is needed to give a "good" representation of a signal. The project also makes significant contact with areas of mathematics outside of harmonic analysis, including ergodic theory and additive combinatorics, and can be expected to stimulate many active discussions with faculty and students of the LSU math department, and with collaborators across the country.