Mathematical Sciences: Harmonic Analysis and Self-Similarity

R. Strichartz将研究自相似测度、分形分析和基本谐波分析估计的实验方法。目标是利用谐波分析的工具发展分形理论，在此过程中丰富这两个领域。通过使中心对象成为一个自相似的度量（在所有尺度上具有相似结构的质量分布）而不是一个集合，考虑分析问题，包括傅里叶变换，卷积算子等，以及纯粹的几何问题变得合理。分析在分形上定义的函数（相对于自相似度量）是近年来非常感兴趣的一个领域。现在可以为这些函数定义傅里叶级数和小波展开的类似物。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。自相似性是一个思想，在应用数学的三个领域中起着重要的作用，这些领域近年来受到了极大的关注：小波、分形和准晶体。但是，自相似的概念——一个物体可以被分解成碎片，每个碎片看起来都像是整体的缩小版——可以用纯数学的方式进行研究，这不仅揭示了它与这些应用的联系，也揭示了它与谐波分析等古老数学领域的联系。R. Strichartz一直走在这些想法发展的前沿，在这个项目中，他将继续这条调查路线。目标是加强和丰富数学基础设施，使数学重要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.