Multilinear and Nonlinear Harmonic Analysis

ch Thiele“多线性和非线性谐波分析”：该项目的科学目的是研究谐波分析中三个相互关联的主题：1)波包分析。从双线性Hilbert变换的有界性证明开始，这类分析得到了广泛的研究，并得出了一系列关于多线性算子和极大算子的结果。2)算术数论，特别是对等差数列、和集和差集的研究。算术数论与上述分析类型之间的联系最近在多线性算子的几个结果中得到了加强。算术数论还与谐波分析中的其他问题有关，例如著名的Kakeya问题。3)非线性傅立叶分析或散射理论。通过散射理论中非线性算子的多线性展开式，将后者与多线性算子联系起来。在过去的三十年里，一维散射理论的代数方面已经得到了广泛的研究，但一些基本的分析问题仍然是开放的，并将在本项目中进行研究。波包分析是数学中的一门学科，任何熟悉乐谱的人都可以很容易地解释它。乐谱是将复杂的音乐作品解码成最基本的部分，即音符，每个音符都用持续时间、音高和音量来描述。波包分析是这方面的数学模拟，它可以用来解码各种各样的数学数据成其基本部分，每一个描述的模拟的持续时间，音高和音量。这种类型的分析对计算机处理来自声学信号、图像、雷达、互联网和其他电信等大量数据的方式产生了巨大的影响。该项目研究与波包分析相关的基本数学问题，这些基础研究不仅可以提高我们对上述过程类型的理解，还可以提高我们对数学中许多不同应用的理解。这个项目也将有助于维持加州大学洛杉矶分校谐波分析研究小组的活跃。

ABSTRACTDMS: 0400879Ch Thiele'Multilinear and Nonlinear Harmonic Analysis':The scientific purpose of this project is to studythree interrelated subjects in harmonic analysis:1) Wave packet analysis. Initiated by the proof ofboundedness of the bilinear Hilbert transform, this typeof analysis has been studied intensely and has led toa series of results on multilinear operators and maximaloperators. 2) Arithmetic number theory, in particularthe study of arithmetic progressions, sum - and difference setsetc. The link between arithmetic number theory and thetype of analysis discussed above has recently be reinforcedbe several results on multilinear operators. Arithmetic numbertheory is also related to other problems in harmonic analysissuch as the famous Kakeya problem.3) Nonlinear Fourier analysis or scattering theory. Viamultilinear expansions of nonlinear operators in scatteringtheory, the latter is linked to multilinear operators.While the algebraic aspects of scattering theory in onedimension have been widely studied over the past thirty years,a number of basic analytic questions remain open and willbe studied in this project.Wave packet analysis is a discipline in mathematicsthat can readily be explained to anyone familiar withmusical scores. A musical score is the decoding of acomplicated musical composition into its most elementaryparts, the notes, each described by duration, pitch, and volume.Wave packet analysis is the mathematical analogue of this,which can be used to decode a large variety of mathematicaldata into its elementary pieces, each described by the analogue of duration, pitch,and volume. This type of analysis has had a tremendous impacton the way computers deal with large sets of data coming fromacoustic signals, images, radar, internet and other telecommunication,etc. This project studies basic mathematical questions associated to wave packet analysis, fundamental research that is likely toimprove our understanding not only of the types of processesdescribed above, but also of many different applications withinMathematics as well. This project will also help to maintain anactive research group in harmonic analysis at UCLA.