基于自仿测度的谱性分析

分形上的Fourier分析是分形几何与Fourier分析相交叉的全新的研究方向。自仿测度的谱性是分形上分析的主要研究内容。本项目以自仿测度为研究对象，对自仿测度的谱性以及它们的应用进行基础研究。具体研究内容：1.高维空间中由实数矩阵和整数数字集生成的自仿测度的谱性，主要考虑无穷正交指数函数存在的充要条件和谱测度的充要条件；2.二维空间上由实数矩阵和整数数字集生成的谱测度的谱特征值问题，例如二维Sierpinski型自仿测度，目的是找到谱测度的谱特征值存在的充要条件。本项目旨在弄清楚越来越多的自仿测度的谱性，并试图为某些经典开问题提供反例，从而丰富了分形几何和调和分析的研究内容。

Fourier analysis on fractal is a new research direction of fractal geometry and Fourier analysis.The spectral properties of self-affine measures are the main content of fractal analysis. In this project, we take the self-affine measures as the research object, and carry out the basic research on the spectral properties of self-affine measures and their applications. The main research contents are as follows: 1. The spectral properties of self-affine measures generated by real matrices and integer digit sets in high-dimensional space. We mainly consider the necessary and sufficient conditions for the existence of infinite orthonormal set of exponential functions and spectral measures. 2. The spectral eigenvalue problem of spectral measures generated by real matrices and integer digit sets in two-dimensional space, such as two-dimensional Sierpinski-type self-affine measures. We want to find the necessary and sufficient conditions for the existence of spectral eigenvalues of spectral measures. This project aims to clarify the spectral properties of more and more self-affine measures, and try to provide counterexamples for some classical open problems, thus enriching the research contents of fractal geometry and harmonic analysis.