Geometric Harmonic Analysis

集合上的调和分析是80年代以来得到广泛发展的一个领域。第一个结果涉及的行为奇异积分集的几何集合。该模型的运营商，并推动理论，是柯西积分与模型的几何设置是一个Lipschitz曲线。这个理论已经见证了爆炸性的增长，在理解L2估计之间的关系，在多尺度几何的基础上设置。20世纪90年代的一个认识是，有一个L2理论的几何在所谓的贝塔数，并有一个“字典”，翻译定理的几何集到定理的小波，反之亦然。虽然贝塔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。正是这样一个理论的发展，沿着它与已经理解的多尺度方面的关系，现在需要提供一个更深入的理解调和分析集。例如，建立局部坐标的问题，捕捉（大多数）低维子集的大部分统计行为还没有得到很好的数学发展，虽然许多建议的方法已经研究。我们建议将自上而下的方法（例如Beta数和相应的几何）与自下而上的扩散几何方法联系起来。 这种研究集合上调和分析的新方法是基于使用与集合有关的某些本征函数。这些本征函数来自自然定义的矩阵，允许通过挑选n个最大的本征值并使用相应的n个本征函数作为坐标来在集合上引入“局部坐标”。所提出的方法与所谓的长轴函数理论有密切关系，因为由此产生的本征函数具有类似的性质。这与Coifman、Jones和Semmes在类似Lipschitz曲线的集合上定义Haar型L2框架的方法形成鲜明对比。该建议的方法给出了不同的功能，人们可以自然地定义局部坐标和研究（近似和正确定义）热流集。琼斯和科夫曼教授建议研究这些方法，并开发一种理论，可以与以前的结果相结合，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的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老问题。这样做的目的是开辟新天地，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.