Takagi函数图像性质的研究

This subject considers Takagi function and it intends to study the integral properties and local structure (including level sets and local level sets) of its graphics. Takagi function is known as a famous Weierstrass-typed fractal function ——continuous but nowhere differentiable, which is induced by the natural combination of fractal geometry, the analysis of wavelet, the Fourier analysis and number theory. It is required that people should have a deeper description on it by the importance appeared on these research area. This subject is intended to study Takagi function by studying the properties of its graphics since lots of properties of a function are reflected by its graphics. However, it is a basic and very difficult problem with which there have not systemic and effective methods to deal, so it is impending and necessary to study. The subject will employ the element tools such as the iterated function systems, Moran set, symbol space and employ the theory of combination, the Schauder equation and Fourier translation. Furthermore, we will study the above problems by using the multi-fractal, the approach by a sequence of simple measures and new number theory and so on. The study of this subject can help people have a deeper understand to the Takagi function and then accelerates the study of fractal functions.

本项目考虑Takagi函数，拟研究其图像的整体性质和局部结构(包含水平集与局部水平集)。Takagi函数是一个著名的Weierstrass型分形函数——连续而处处不可微，它是分形几何、小波分析、Fourier分析和函数论自然结合的产物，它在这些相关学科显示的重要性要求人们对其有更深刻的认识。因为函数的很多性质能够通过其图像反映出来，所以本项目拟通过了解图像的性质来认识Takagi函数。但这方面的研究目前还没有系统和行之有效的方法，它是一个基本而非常困难的问题，因此对其进行研究就显得非常迫切和必要了。本项目拟采用分形理论中的基本工具(如迭代函数系、Moran集、符号空间等)，结合组合理论、Schauder方程、Fourier变换和加细方程，运用重分形分析、简单测度卷积列逼近、某些新的数论等方法研究上述问题。本项目的研究有助于人们进一步认识Takagi函数，从而促进分形函数的研究和发展。

本项目按申请计划如期进行，研究了Takagi函数局部水平集的推广——LM局部集的几何性质。采用分形理论中的迭代函数系、Moran集、符号空间等基本手段，结合组合理论等方法，给出了[0,1]区间内的任意一点所对应的LM局部集的Hausdorff维数，证明了LM局部集的Hausdorff维数大于零的点构成的集合的Hausdorff维数为1。这也确定了Takagi函数所有局部水平集的Hausdorff维数。本项目结果有助于人们进一步认识Takagi函数。