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立叶分析及其在完全可积系统中的应用有关的几个问题。这个项目是关于傅里叶级数沿着部分和的‘瘦’子序列的收敛，其动机是Kanyagin的一个猜想；与傅里叶级数有关的经典和非经典估计的推广，其动机来自于偏微分方程中Germain-Masmudi-Shatah的工作；具有粗糙变权的正交多项式的Plancherel-Rotach渐近性的一种新方法，其动机是在酉随机矩阵的谱研究中的应用以及在其他非交换的傅立叶变换例子中的应用；以及Ablowitz-Kaup-Newell-Segur可积方程长时间渐近中低阶项的精确估计，另外的动机来自对可积方程的摄动的研究。这个数学研究项目是在调和分析领域，在这个领域中，通过将数据分解为更易于管理的分量来分析数据。一个重要的工具是傅里叶变换，它提供了将复杂的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图像结构分析中是有用的。更广泛地说，可以在信号处理中使用谐波分析来压缩和解压缩数据，从而允许在错误可控的情况下有效地传输大量信息。傅里叶变换的其他变体也出现在统计学和数学物理中。一个这样的变种出现在随机物理系统的能级研究中。本研究项目旨在通过研究傅里叶变换的新变种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.