Fourier Analysis: Old Themes, New Perspectives

傅里叶分析：老主题，新视角傅里叶分析中的许多问题与这门学科本身一样古老。为了理解其中的一些问题，这项研究取得了非凡的进展，发现了与许多其他科学领域的新联系，揭示了新的观点。这些新观点之一是多元线性观点，它极大地改进和丰富了我们通过冻结变量方法研究问题的经典方法。多线性分析的基本思想是将线性无关的泛函参数作为线性变量来处理，从而获得更高的自由度和更大的灵活性，这往往会导致更深入、更深远的理解。首席研究员建议着手对平移不变的多线性算子进行广泛的研究，既要广泛到足以涵盖大多数已知的例子，又要深入到足以包括非常奇异的特殊算子。本文由实证理论、实证结果和一般理论三部分组成。关于正理论的研究工作包括双线性Hilbert变换和其他多线性粗糙奇异积分的有界范围的扩展。否定结果集中在局部平方可积情况外的圆盘乘子、可积函数积上的双线性希尔伯特变换、近可积函数空间上的Carleson-Hunt算子以及上述所有算子的某些高维版本上。提出的一般理论将侧重于有关多线性插值和外推的主题。这些项目提供了简化本学科研究的重要工具。傅里叶分析的主要目的是通过分解成更简单的部分来研究复杂的对象。就像音乐可以分解成一个简单基本频率的整数倍的组合一样，复杂的算子可以用它们在频谱上的作用来表示。傅里叶分析提供了将同一函数（信号）的特殊（时间）和相位（频率）方面联系起来的工具。一旦将信号和图像分解成小块并通过傅里叶分析进行研究，就可以更好地定位信号和图像的不规则性。例如，噪声和模糊很容易定位与傅里叶变换的应用，但现在甚至更具有挑战性的壮举可以实现。在本研究中，将使用对时间和频率考虑都敏感的分解技术来研究某些算子。这种运算符通常是通过通过乘以一个固定的、通常是非平滑的函数来改变输入信号的频率来定义的。实际上，由于气象现象而使无线电通信或电视传输突然中断就是这种操作员的例子。防止信息丢失的保护可以在数学上建模为可积性的保存，这里提出对其进行研究。

DMS 0400387PI: Loukas GrafakosUnviersity of Missouri Fourier Analysis: Old themes, new perspectiveAbstractMany problems in Fourier Analysis are as old as the subject itself.The quest to understand some of these problems has pushed the research to such extraordinary developmentsthat new connections with many other scientific areashave been discovered andnew perspectives revealed. One of these new perspectivesis the multilinear point of view that has significantly refined and enriched our classical approach of studying problems via the method of freezing variables. The basic idea in multilinear analysis is that linearly independent functional parameters are treated as linear variables, allowing a higher degree of freedom and greater flexibility, which often results in a deeper and more far-reaching understanding.The principal investigator proposes to embark on an extensive study of translation-invariant multilinear operators, both broad enough to cover most known examples, but also deep enough to include special operators of a very singular nature. This proposal consists of three parts: positive theory, negative results, and general theory. Proposed work on the positive theory includes extension of the range of boundedness of the bilinear Hilbert transform and other multilinear rough singular integrals. Negative results focus on the disc multiplier outside the locally square integrable case, the bilinear Hilbert transform on products of integrable functions, as well as the Carleson-Hunt operator on spaces of nearly integrable functions and certain higher dimensional versions of all the previous operators.The proposed general theory will focus on topics concerningmultilinear interpolation and extrapolation.These items provide important tools that simplify the study of the subject.The main goal of Fourier Analysis is to study complicated objects via decompositions into simpler pieces. Just as music can be disassembled into compositions of integral multiples of a simple basic frequency, complicated operators can be represented by their actions on a spectrum of frequencies. Fourier analysis provides the tools that relate special (time) and phasial (frequency) aspects of the same function (signal).Irregularities of signals and images are better located once these are decomposed into small pieces and studied via Fourier analysis. For instance, noise and blurring are easily located with the application of the Fourier transform, but nowadays even more challenging feats can be achieved. In this investigation, certain operators are to be studied using decomposition techniques sensitive toboth time and frequency considerations. Such operators are often defined by altering the frequency of input signals via multiplication by a fixed and often nonsmooth function. In practice, the abrupt interruption of radio communication or television transmission by a meteorological phenomenon are examples of such operators.The protection against the loss of information can bemathematically modeled as the preservation of integrability and this is proposed to be investigated here.