自仿集合在小波分析和Fuglede谱集猜想中的应用

The study in this research mainly concentrate on the applications of self-affine sets to wavelet theory and Fuglede's spectral set conjecture. The main contents are as follows: (1) In view of the connection between the theory of integral self-affine tiles and wavelet analysis, we will consider how to construct wavelet sets using general self-affine tiles and characterize the properties of the corresponding dimension functions. (2) We will construct the relationship between the pseudo Hausdorff measure of self-affine set and the upper Beurling density of the infinite discrete set iterated by the digit set. (3) In dimension one, we will consider if a self affine tile is a spectral set with the help of the structure of its digit set. In higher dimensions, the spectral property of a special class of self-affine tiles will be studied. (4) We will study the spectral problem of the self-affine measure determined by a more general digit set and characterize how a digit set determines the spectral property of the corresponding self-affine measure.. We hope our research could provide a profound knowledge of the relationship among the digit sets, the tiling property and the spectral property of self-affine measure. Furthermore, we hope this research will lay the foundation for solving Fuglede's spectral set conjecture, the representation of Hausdorff measure of self-affine sets, the frame theory of self-affine measures.

本项目主要研究自仿集合在小波分析理论和Fuglede谱集猜想中的应用。具体包括：(1) 我们将借助自仿整tile理论和小波分析之间的关系来考虑一般自仿tile构造小波集合的问题，并刻画相应的尺度函数的性质。(2) 建立自仿集合的伪Hausdorff测度和由其数字集迭代生成的无穷离散集的上Beurling密度之间的关系。(3) 在一维欧式空间上，通过自仿tile所对应的数字集的结构特征来研究自仿tile是否是谱集合；同时考虑高维上一类特殊自仿tile的谱问题。(4) 研究一般数字集所决定的自仿测度的谱性质，并且刻画数字集和相应自仿测度谱性质之间的关系。. 我们希望通过本项目的研究，对数字集和tiling性质，内部结构以及谱性质之间的关系有更加深刻的认识。为Fuglede谱集猜想，自仿集的Hausdorff测度的一般表示，自仿测度的标架理论打下基础。

在本项目的支持下，我们在傅里叶分析和分形理论的框架下主要的研究如下：(1) 鉴于集合的拓扑结构和谱集猜想及小波分析的关系，我们研究了整自仿multi-tiles的表示;(2)对于一般的自仿集合，我们给出了其对应的迭代函数系满足开集条件，且建立了自仿集合的伪Hausdorff测度和由其数字集迭代生成的无穷离散集的上Beurling密度之间的关系；(3)通过自仿tile的数字集结构开启了自仿tile的谱性质研究，证明了一类自相似tile是谱集合及谱集猜想在一些特殊情形下是成立的;(4)考虑了奇异测度和的平移绝对连续性以及其上的傅里叶标架存在性问题。通过此项目的研究，我们数字集和谱性质之间的关系有了更加深刻的理解和认识。为Fuglede谱集猜想，自仿集的Hausdorff测度的一般表示，自仿测度的标架理论打下基础。