Fourier analysis and applications to completely integrable systems

做和他的合作者将使用新的方法来调查几个问题有关的傅立叶分析及其应用完全可积系统。这个项目涉及傅里叶级数沿着部分和的沿着“薄”子序列的收敛，其动机是Konyagin的一个猜想;将与傅里叶级数有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;给出了一种新的粗变权正交多项式的Plancherel-Rotach渐近性的研究方法，由酉随机矩阵的谱研究中的应用和傅里叶变换的其他非交换例子中的应用所激发;以及对Ablowitz-Kaup-Newell-Segur可积方程长时间渐近性中低阶项的精确估计，以及来自可积方程扰动研究的额外动机。 这个数学研究项目是在谐波分析领域，在该领域中，通过将数据分解为更易于管理的组件来分析数据。一个重要的工具是傅里叶变换，它提供了将复杂信号分解为基本波的方法。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，谐波分析在去噪图像和分析图像结构中是有用的。更一般地说，谐波分析可以用于信号处理以压缩和消除数据，从而允许以可管理的错误有效地传输非常大量的信息。 傅里叶变换的其他变体也出现在统计学和数学物理学中。一个这样的变体发生在随机物理系统的能级研究中。本研究项目旨在通过调查傅立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.