Mathematical Sciences: Harmonic Analysis and Self-Similarity

本项目旨在探索谐波分析中一些发展中的思想，这些思想与自相似性中的重要思想有关。由于自相似性与分形几何和小波分析的关系，它是当今人们非常感兴趣的一个主题。然而，这些关系可以追溯到30年代早期Wiener和Wintner对康托测度的傅里叶变换的研究。要做的工作包括计算自相似测度类的维数。这一概念将向两个方向扩展。第一种方法是用平均收缩性取代严格收缩变换的定义。在这种情况下，不能再保证紧凑的支持。这些度量的增长率和它们的傅立叶变换的平滑性是研究的主题。第二个方向用可变权值替换凸组合。在这里，我们感兴趣的是度量的存在性和唯一性，以及它们的Lp和点维。对自相似测度的Radon变换进行了分析。要考虑的两个自然问题是Radon变换是否可逆以及是否可以表征变换的范围。当且仅当测度的变换不缓慢减小时，已知变换能保持这类分布。对于自相似度量，情况并非如此，留下了适当空间应该是什么的问题。谐波分析结合了最能体现综合思想的数学元素。人们试图把复杂的问题分解成基本的组成部分。然后分析这些组件的基本特性。最后，通过组件的重组重构解。傅里叶级数和傅里叶变换是在这种情况下使用的工具的例子；一个是离散的，另一个是连续的。最近，小波理论为一些更经典的谐波分析方法增加了新的维度。

This project seeks to exploit some of the developing ideas in harmonic analysis which relate to important ideas in self-similarity. Self-similarity is a subject of intense interest today because of its relationship with fractal geometry and wavelet analysis. However, these relationships can be traced to work in the early 30's by Wiener and Wintner on Fourier transforms of Cantor measures. Work to be done includes the computation of dimensions of classes of self-similar measures. There are two directions in which the concept will be expanded. The first involves replacing strictly contractive transformations in their definition and replacing this with average contractivity. One can no longer guarantee compact support in this case. The growth rate of such measures and the smoothness of their Fourier transform are subjects for investigation. The second direction replaces the convex combinations with variable weights. Here one is interested in existence and uniqueness of the measures as well as their Lp and pointwise dimensions. Work will also be done in analyzing the Radon transform of self-similar measures. The two natural questions to be considered are whether or not the Radon transform is invertible and whether one can characterize the range of the transform. The transform is known to preserve the class of distributions if and only if the transform of the measure is not slowly decreasing. For self-similar measures this is not the case, leaving open the question of what the proper spaces ought to be. Harmonic analysis combines those elements of mathematics best exemplifying the ideas of synthesis. One seeks to decompose complex problems into fundamental components. These components are then analyzed for their basic characteristics. Finally, the solution is reconstructed through a recombination of the components. The Fourier series and Fourier transform are examples of tools used in this context; one discrete , the other representing a continuous decomposition. More recently the wavelet theory added new dimensions to some of the more classical approaches to harmonic analysis.