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Symmetry Parameter Analysis of Singular Integrals

奇异积分的对称参数分析

基本信息

批准号：
2054863
负责人：
Brett Wick
金额：
$ 19.76万
依托单位：
Washington University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054863&HistoricalAwards=false
关键词：
Symmetry Parameter Analysis Singular Integrals

项目摘要

Harmonic Analysis is the branch of mathematics concerned with the rigorous description of signals (functions) and of their processing (operators). Examples of signals are sound, images, time series and weather data. Such signals are analyzed via the overlaying (superposition) of basic harmonics of well-specified duration, intensity and frequency. These basic harmonics are functions called wavelets. Image or audio denoising, compression, or pattern recognition are accomplished by filter processing, which refers to a suitable superposition of each wavelet after the action of the filter on it. This is also known as the time-frequency method. A particular concrete example of a re-construction process is used in tomographic imaging, where the shape of a solid body is re-composed from samples of the body along one or two-dimensional rays of penetrating waves, which can be mathematically described as lines or planes in three dimensional space. One component of this mathematics research project focuses on a new family of methods for the wavelet description of the class of singular integral operators, arising for instance in the time-frequency analysis of highly oscillatory signals. Another component of this research project is concerned with the mathematical properties of sampling solid objects along lines or planes. The integrated broader impact activities focus on strengthening the pool of socioeconomically disadvantaged, ethnical minority students (underrepresented groups) in graduate degrees in mathematics and improving retention. Activities connected to training and mentoring of graduate students in Analysis and topical dissemination of knowledge will also be carried out. The broad aim of the first circle of questions is to produce representation formulas for classes of singular integrals in terms of so-called model operators conserving the same invariance structure. This paradigm applies to Zygmund-type operators and modulation invariant operators akin to the bilinear Hilbert transform, both of which are out of reach for dyadic-probabilistic methods. Concrete applications come from elliptic and dispersive PDE, operator theory and quasi-conformal mappings. The second related family of questions is motivated by pointwise convergence of bilinear ergodic averages for Banach-valued functions, a celebrated theorem by Bourgain in the scalar case. The approach is based on Banach-valued variational estimates for the truncated bilinear Hilbert transform. The central item in a further set of questions concerning directional singular integrals is a version of the Kakeya maximal estimate where tubular averages are replaced with averages over singular line segments, and more generally, n-dimensional subspaces. One source of motivation is the connection with Fourier restriction in higher codimensions. The methods involve algebra-geometric techniques such as polynomial partitioning on manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

调和分析是数学的一个分支，涉及对信号(函数)及其处理(运算符)的严格描述。信号的例子有声音、图像、时间序列和天气数据。这些信号通过特定持续时间、强度和频率的基波叠加(叠加)来分析。这些基本的谐波是被称为小波的函数。图像或音频的去噪、压缩或模式识别是通过滤波处理来完成的，滤波处理指的是在滤波器作用于每个小波之后每个小波的适当叠加。这也被称为时频方法。重建过程的特定具体示例用于断层成像，其中实体的形状由沿穿透波的一维或二维射线的实体样本重新组成，其可以在数学上描述为三维空间中的线或平面。这个数学研究项目的一个组成部分集中于一类奇异积分算子的小波描述的新方法，例如在高振荡信号的时频分析中出现的方法。这项研究项目的另一个组成部分是关于沿线或面采样实体对象的数学特性。综合的、更广泛的影响活动侧重于加强在社会经济上处于不利地位的少数族裔学生(代表人数不足的群体)在数学研究生学位中的地位，并提高保留率。还将开展与研究生在分析和专题知识传播方面的培训和辅导有关的活动。第一个问题圈的主要目的是在保持相同不变结构的情况下，用所谓的模型算子给出奇异积分类的表示公式。这一范例适用于Zygmund型算子和类似于双线性Hilbert变换的调制不变算子，这两者都不是并矢概率方法所能达到的。具体的应用来自于椭圆型和色散型偏微分方程组、算子理论和拟共形映射。第二类相关问题的动机是Banach值函数的双线性遍历平均的逐点收敛，这是Bourain在标量情况下的著名定理。该方法基于截断双线性Hilbert变换的Banach值变分估计。在另一组关于方向奇异积分的问题中，中心项是Kakeya最大估计的一个版本，其中管状平均被奇异线段上的平均替换为更一般地，n维子空间上的平均。动机的一个来源是与高余维中的傅立叶限制的联系。这些方法涉及代数几何技术，如流形上的多项式划分。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。