Mathematical Sciences: Harmonic Analysis and Self-Similarity

Strichartz 9623250 R.Strichartz将研究与自相似度量、分形学分析以及基本调和分析估计的实验方法相关的问题。目标是利用调和分析工具发展分形学理论，在这个过程中丰富这两个领域。通过使中心对象成为自相似度量(在所有尺度上具有相似结构的质量分布)而不是集合，可以考虑涉及傅立叶变换、卷积算子等的解析问题，以及纯粹的几何问题。对定义在分形图上的函数的分析(关于自相似度量)是近年来非常感兴趣的一个领域。现在可以为这类函数定义傅里叶级数和小波展开的类似。本项目将扩展这一理论，并回答有关这些扩展的一些基本问题。自相似是一个在应用数学的三个领域发挥重要作用的概念，这三个领域近年来受到了极大的关注：小波、分形学和准晶。但自相似的概念--一个可以被分解成碎片的物体，每个碎片看起来都像整体的缩小版--可以用纯数学的方式来研究，它不仅与这些应用程序有关，而且与调和分析等更古老的数学领域也有联系。R.Strichartz一直走在这些想法发展的前沿，并将在这个项目中继续这一调查。其目标是加强和丰富数学基础设施，使重要数学应用的快速发展成为可能。

Abstract Strichartz 9623250 R. Strichartz will investigate problems related to self-similar measures, analysis on fractals, and an experimental approach to basic harmonic analysis estimates. The goal is to develop the theory of fractals using the tools of harmonic analysis, in the process enriching both areas. By making the central object a self-similar measure (a mass distribution that has similar structure at all scales) rather than a set, it becomes plausible to consider analytic questions, involving Fourier transforms, convolution operators, etc., as well as purely geometric questions. Analysis of functions defined on fractals (with respect to a self-similar measure) is an area of great interest in recent years. It is now possible to define the analogs of Fourier series and wavelet expansions for such functions. This project will extend this theory and answer some basic questions about these expansions. Self-similiarity is an idea that plays an important role in three areas of applied mathematics that have received a great deal of attention in recent years: wavelets, fractals, and quasicrystals. But the idea of self-similarity -- an object that can be broken up into pieces, each of which looks like a scaled down version of the whole -- can be investigated in a purely mathematical way that brings out its connections not only with these applications but also older areas of mathematics such as harmonic analysis. R. Strichartz has been in the forefront of the development of these ideas, and in this project will continue this line of investigation. The goal is to strengthen and enrich the mathematical infrastructure that makes possible the rapid development of important applications of mathematics.