Mathematical Sciences: Discrete Littlewood-Paley Theory

经典谐波分析的三个主题将是 在这个数学研究项目中发展起来的。 存在的问题 研究的中心是离散分解技术， 最近的发展。 第一个领域将集中在Littlewood-Paley理论。 一 傅立叶级数基本思想的推广 哪些函数或分布被表示为组合 简单的功能。 从源函数到 相应的系数序列称为φ- 傅里叶变换也许是最著名的 example. 这项研究的一个应用将是开发 推广大卫和柯勒律纳的T（1）定理的一个判据 给出了奇异线性系统的精确有界性条件 运算符仅仅基于它们对常数的作用 功能协调发展的 其他工作将发展密切联系， Littlewood-Paley理论和加权范数不等式， 特别是涉及用于类的因式分解和外推， 权重远大于传统的Ap权重。 之一 这些研究的动机来自于寻找尖锐的 傅里叶变换的权值有界的条件。 为了研究这样的不等式，我们必须在频率上局部化， 时间和时间同时进行。 Littlewood-Paley理论 设计Φ变换来处理这样的问题。 离散Littlewood-Paley理论已被证明是非常 可用于地震和声学分析，心理生理学， 计算机视觉和图像处理以及量子领域 理论 大部分的算法开发计算工作 在这些领域需要采样技术。 这项工作将 设法发展有效取样过程的特征 与这些算法有关。

Three themes of classical harmonic analysis will be developed in this mathematical research project. The problems to be investigated center on discrete decomposition techniques under recent development. The first area will focus on Littlewood-Paley theory. A generalization of the fundamental ideas of Fourier series in which functions or distributions are represented as combinations of simple functions. The map from the source function to the corresponding sequence of coefficients is called the phi- transform, the Fourier transform being perhaps the best known example. One application of this study will be to develop criteria for an extension of the T(1) theorem of David and Journe which gives precise boundedness conditions for singular linear operators solely on the basis of their action on constant functions. Other work will develop the close connection between Littlewood-Paley theory and weighted norm inequalities, especially to factorization and extrapolation for classes of weights much larger than the traditional Ap weights. One of the motivations for such studies comes from efforts to find sharp conditions on weights for the Fourier transform to be bounded. To study such inequalities, one must localize on the frequency side and time side simultaneously. Littlewood-Paley theory and the phi-transform are designed to deal with such problems. Discrete Littlewood-Paley theory has proved to be very useful in seismic and acoustic analysis, psycho-physiology, computer vision and image processing as well as in quantum field theory. Much of the algorithm development for computational work in these areas requires sampling techniques. This work will seek to develop characterizations of efficient sampling processes associated with these algorithms.