Fourier analysis and applications to completely integrable systems

Do和他的合作者将使用新的方法来研究与傅里叶分析及其在完全可积系统中的应用有关的几个问题。这个项目关注傅立叶级数沿部分和的“细”子序列的收敛性，由Konyagin的一个猜想激发；将傅立叶级数相关的经典和非经典估计扩展到加权设置，其动机来自Germain-Masmoudi-Shatah在偏微分方程中的工作；粗糙变权正交多项式的Plancherel-Rotach渐近性的一种新方法，在酉随机矩阵的谱研究和其他傅里叶变换的非交换例子中的应用的推动下；以及ablowitz - kap - newwell - segur可积方程长期渐近中低阶项的尖锐估计，并有来自可积方程摄动研究的额外动机。这个数学研究项目是在谐波分析领域，在谐波分析中，通过将数据分解成更易于管理的成分来分析数据。傅里叶变换是一个重要的工具，它提供了将复杂信号分解成基本波的方法。这些分解和相关技术在工程和其他应用科学中有许多有用的应用。例如，在图像处理中，谐波分析在图像去噪和分析图像结构方面很有用。更普遍的是，谐波分析可以用于信号处理，以压缩和解压数据，允许非常大量的信息以可控的误差有效地传输。傅里叶变换的其他变体也出现在统计学和数学物理中。其中一种变体出现在对随机物理系统的能级的研究中。本研究项目旨在通过研究傅里叶变换的新变体的基本性质来提高对上述研究的理解。

Do and his collaborators will use new methods to investigate several questions related to Fourier analysis and its applications to completely integrable systems. This project concerns convergence of Fourier series along `thin' sub-sequences of partial sums, as motivated by a conjecture of Konyagin; extensions of classical and non-classical estimates related to Fourier series to weighted settings, with motivations coming from work of Germain-Masmoudi-Shatah in partial differential equations; a new approach to Plancherel-Rotach asymptotics of orthogonal polynomials with rough varying weights, motivated by applications in spectral study of unitary random matrices and by applications in other non-commutative examples of the Fourier transform; and sharp estimates for lower-order terms in long-time asymptotics of Ablowitz-Kaup-Newell-Segur integrable equations, with additional motivations coming from studies of perturbation of integrable equations. This mathematics research project is in the area of harmonic analysis, in which data is analyzed by breaking it into more manageable components. An important tool is the Fourier transform, which provides methods to decompose complicated signals into basic waves. These decompositions and related techniques have a number of useful applications in engineering and other applied sciences. For example, in image processing, harmonic analysis is useful in de-noising images and analyzing image structures. More generally, harmonic analysis could be used in signal processing to compress and decompress data, allowing for very large amounts of information to be effectively transmitted with manageable errors. Other variants of the Fourier transform also appear in statistics and mathematical physics. One such variant occurs in the study of energy levels of random physical systems. This research project seeks to improve the understanding of the above studies by investigating fundamental properties of the new variants of the Fourier transform.