分形谱测度及相关问题研究

The study of spectral measures on Euclidean space dates back to the famous spectral conjecture raised by Fuglede in 1974. In this project, by developing the existing tools and techniques and introducing the knowledge of tree structure, spectral maximal mapping in higher dimension, Carleson's operator theory, we mainly intend to carry out the study on the following three aspects: (1) The investigation on spectrality and non-spectrality of random convolutions on Euclidean space. We will investigate the relationship between the digit sets and the spectrality of random convolutions, and characterize the relationship between the maximal orthogonal sets and spectrality by building the tree structure and defining the maximal spectral mapping. (2) The investigation on the spectral eigen-matrix problem of random convolutions on Euclidean space. Following on the former reserach [20,21,22], we will determine the spectral eigen-matrix and spectral eigen-spectra of spectral random convolutions which satisfy common compatible pairs. (3) The study on the convergence and divergence of Mock Fourier series of p-integrable functions on 4-Cantor measure. We will investigate this by building Carleson's operator on fractal spaces.

欧氏空间上谱测度的广泛研究起源于Fuglede在1974年提出的著名的谱集猜测。在本项目中，我们将在发展现有工具的基础上，引入高维空间中的树结构、极大谱映射、Carleson算子理论知识等来研究分形谱测度以及相关问题。本项目将重点研究如下内容：(1)高维欧氏空间中随机卷积的谱性与非谱性质。我们将深入探究高维欧氏空间中由特殊数字集生成的随机卷积测度的谱性，并定义树结构及极大谱映射来刻画其极大正交集与谱性的关系。(2)高维欧氏空间中随机卷积的谱特征矩阵问题。我们将在前期工作基础[20,21,22]上深入探究高维欧氏空间中满足公共相容对条件的随机卷积谱测度的谱特征矩阵及其特征谱。(3)四分Cantor集上p次可积函数的Mock Fourier级数的点态敛散性研究。我们拟建立分形测度函数空间上的Carleson算子理论探究其Mock Fourier级数的敛散性。

欧氏空间上谱测度的广泛研究起源于Fuglede在1974年提出的著名的谱集猜测。本项目共完成6篇学术论文，其研究内容及重要结果主要集中在以下方面 （1）证明得到由缺项序列及相容对生成的实直线上的齐次Moran测度必定是谱测度，而高维空间中由缺项序列生成的齐次Moran谱测度并不一定由相容对生成；（2）系统刻画出由有限多个三元素数字集生成的随机卷积测度的极大正交集，并给出其成为谱的充分条件及必要条件；（3）系统研究广义伯努利卷积测度(含四分康托尔测度)上mock傅里叶级数的收敛性问题，证明给出存在不可数多个谱使得其mock傅里叶级数同时满足一致收敛、L^p收敛及点态收敛。谱测度分析是当前国际数学领域最热门的研究分支之一，本项目的研究结果将在一定程度上促进分形谱测度理论及相关领域的发展。