Geometric Harmonic Analysis

集合上的调和分析是自20世纪80年代S以来一直在深入发展的一个领域。第一个结果是关于集合上的奇异积分在集合几何方面的行为。模型的运算符和理论的推动力是柯西积分，模型的几何设置是一条Lipschitz曲线。这一理论在理解基础集合的多尺度几何方面的L2估计之间的关系方面取得了爆炸性的增长。90年代S的一项认识是，存在以所谓Beta数表示的L2几何理论，并且有一本将集合几何定理翻译成小波定理的“词典”，反之亦然。虽然Beta数理论也很好地理解了数据集的多尺度结构，但它只为理解几何结构提供了一定的框架，并不包含类似于傅立叶级数或热流研究的理论。正是这种理论的发展，以及它与已经理解的多尺度方面的关系，现在需要提供对集合上的调和分析的更深层次的理解。例如，建立局部坐标来捕捉(主要)低维子集的大部分统计行为的问题在数学上还没有得到很好的发展，尽管已经有许多提出的方法被研究。我们建议将自上而下的方法(例如Beta数和相应的几何)与扩散几何的自下而上方法联系起来。这种研究集合调和分析的新方法是基于与集合相关的某些特征函数的使用。这些特征函数来自自然定义的矩阵，通过选取n个最大的特征值并使用相应的n个特征函数作为坐标，允许在集合上引入“局部坐标”。所提出的方法与所谓的长函数理论有着密切的关系，因为所得到的特征函数具有相似的性质。这与Coifman、Jones和Semmes在类似Lipschitz曲线的集合上定义Haar型L2框架的方法形成了鲜明的对比。提出的方法给出了不同的函数，可以自然地定义局部坐标和研究(近似和正确定义的)集合上的热流。琼斯教授和科夫曼教授建议研究这些方法，并开发一种理论，可以与之前的结果相结合，将自上而下的行为与自下而上的行为联系起来。这是从计算效率和快速算法发展的角度来完成的。他们还建议研究各种几何描述，以提供新的方法来解决调和分析中的旧问题。通过这样做，他们的目标是开辟新的道路，并扩大其他早期方法的适用性。

Harmonic analysis on sets is an area that has been under intensive developmentsince the 1980's. The first results concerned the behavior of singular integrals on sets in terms of the geometry of the sets. The model operator, and impetus to the theory, was the Cauchy integral with the model geometric setting being a Lipschitz curve. This theory has witnessed an explosive growth in terms of understanding the relation between L2 estimates in terms of the multiscale geometry of the underlying set. One of the realizations of the 1990's was that there is an L2 theory of geometry in terms of so-called Beta numbers, and that there is a "dictionary" that translates theorems on geometry of sets into theorems on wavelets, and vice versa. While the theory of Beta numbers also gave a good understanding of the multiscale structure of e.g. a data set, it only provides a certain framework for the understanding of the geometry, and does not encompass a theory analogous to Fourier series or the study of heat flow. It is the development of such a theory, along with its relations to the already understood multiscale aspects, that is now required to provide a deeper understanding of harmonic analysis on sets. For example the problem of building local coordinates that capture most of the statistical behavior of (mostly) lower dimensional subsets has not been well developed mathematically, though many proposed methods have been studied. We propose to relate the top down methods (e.g. Beta Numbers and corresponding geometry) to bottom up methods of diffusion geometries. This new method of studying harmonic analysis on sets is based on the use of certain eigenfunctions related to the set. These eigenfunctions, coming from naturally defined matrices, allow the introduction of "local coordinates" on the set by picking the n largest eigenvalues, and using the corresponding n eigenfunctions as coordinates. The method proposed has a close relation to the theory of so-called prolate functions, as the resulting eigenfunctions have similar properties. This is in sharp contrast to the method of Coifman, Jones, and Semmes for defining Haar type L2 frames on sets resembling Lipschitz curves. The method of the proposal gives different functions with which one can naturally define local coordinates and study (approximate and correctly defined) heat flow on sets. Professors Jones and Coifman propose to study these methods and develop a theory that can be combined with previous results to relate top down behavior to bottom up behavior. This is done from the point of view of computational efficiency and the development of fast algorithms. They also propose to study the various geometrical descriptions to provide new methods of attacking older problems in harmonic analysis. In doing so, they aim to break new ground and broaden the applicability of other earlier methods.