Problems in Time Frequency Analysis

时频分析中的问题

• 批准号: 1161752
• 负责人: Ciprian Demeter
• 金额: $ 21.45万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1161752&HistoricalAwards=false
• 关键词: Problems; Time; Frequency; Analysis

This mathematics research project aims at answering questions in harmonic analysis that are of great interest in modern analysis. First, Demeter proposes to investigate the fundamental problem of whether the phase space translates of various classes of functions are linearly independent. Together with his collaborators, Demeter has recently introduced two new tool: One is the spectral theory of random Schroedinger operators, the other one is the theory of simultaneous Diophantine approximation. It is expected that further analysis will reveal connections with various other fields of mathematics. Second, the question of differentiability and boundedness of singular integrals along vector fields has gained a huge momentum over the last five years. Building on his recent work, Demeter proposes to use analytic and combinatorial methods to investigate various Kakeya-like maximal operators, and to obtain optimal bounds for the Hilbert transform along a given number of directions in the plane. Demeter will also investigate the bilinear Fourier restrictions to domains with curvature. Among other things, this involves extending known uniform estimates for the bilinear Hilbert transform. This research project will employ a wide variety of tools from diverse areas of mathematics such as harmonic analysis, combinatorics, number theory, spectral theory and ergodic theory.The publication of the work that is expected to arise from this mathematics research project will enhance the mathematical community's understanding of the important connections among various mathematics research areas, such as harmonic analysis, number theory and ergodic theory. The resolution of the Heil-Ramanathan-Topiwala conjecture could have a significant impact on various sciences, such as signal processing, where Gabor and wavelet analysis play an important role. This research project includes a human resources component consisting of training graduate students and postdoctoral researchers.

这个数学研究项目旨在回答调和分析中现代分析中非常感兴趣的问题。首先，Demeter提出了一个基本问题，即各种函数的相空间平移是否线性无关。Demeter最近和他的合作者一起推出了两个新工具：一个是随机薛定谔算子的谱理论，另一个是同时丢芬图近似理论。预计进一步的分析将揭示它与其他数学领域的联系。其次，奇异积分沿向量场的可微性和有界性问题在过去五年中获得了巨大的发展势头。在他最近工作的基础上，Demeter建议使用分析和组合方法来研究各种类kakeya极大算子，并在平面上沿给定数量的方向获得希尔伯特变换的最优界。Demeter也将研究双线性傅立叶限制对曲率域的影响。除其他事项外，这涉及扩展已知的双线性希尔伯特变换的一致估计。这个研究项目将使用各种各样的工具，从不同的数学领域，如谐波分析，组合，数论，谱理论和遍历理论。该数学研究项目预计将产生的成果的出版将提高数学界对各种数学研究领域之间重要联系的理解，如谐波分析、数论和遍历理论。Heil-Ramanathan-Topiwala猜想的解决可能会对各种科学产生重大影响，例如信号处理，其中Gabor和小波分析发挥着重要作用。该研究项目包括人力资源部分，包括培养研究生和博士后研究人员。