【作者】 丁道新；

【导师】 文志雄；

【作者基本信息】 华中科技大学 ， 理论物理， 2010， 博士

【摘要】 分形与框架的相关问题是分形几何中有趣而重要的课题.本论文研究了分形级数,刻画了Fourier框架与Tilings的一种关系,并且利用迭代函数系统构造了一类广义连续框架.本论文总共有三部分.第一部分我们描述了一种自然的方法利用d-维欧氏空间中的绝对收敛级数来构造分形集,并对这类集合的结构特点作出了一些描述.然后,对一维直线上的情况,我们给出两个条件分别得到了区间和齐次Moran集.最后,对高维的情况,我们给出了一个充分条件来计算具有重叠结构的集合的Hausdorff维数,并且给出了一些例子.第二部分我们首先给出了一个可分Hilbert空间的广义连续框架的概念.在得出广义连续框架的一些基本结果之后,我们利用迭代函数系统、概率函数以及窗口函数构造了一类广义连续框架.论文的最后一部分介绍了谱集猜想.在研究谱集猜想的过程中,Jorgensen和Pedersen提出了一个猜想.受此驱动,Lagarias、Reeds和Wang得到了谱与Tilings的一个关系刻画,而且结合Keller标准就能证明Jorgensen和Pedersen猜想.Iosevich、Pedersen和Kolountzakis利用另外的技巧也证明了Jorgensen和Pedersen猜想.在本文中,我们考虑Fourier框架并得到了Fourier框架与Tilings的一个关系,由此刻画了谱与Tilings的一个更一般的关系.更多还原

【Abstract】 The related problems of fractals and frames are both important and interesting in fractal geometry. In this thesis, we study the fractal series, obtain some characterizations of Fourier frames and tilings, and construct a class of generalized continuous frames by using iterated function systems.This thesis contains three parts.In the first part of this thesis, we describe a natural way to associate fractals to a certain class of absolutely convergent series in Rd, and portray the structures of the sets. In the case for d= 1, we give two sufficient conditions for the set associated to series being an interval or a homogeneous Moran set, respectively. Finally, for d≥1, we give a sufficient condition to calculate the Hausdorff dimension of some fractals with overlapping, and present some examples.In the second part of this thesis, we give the definition of a generalized continuous frame for a separable Hilbert space. For such a generalization, after obtaining some basic results about these frames, we construct a class of generalized continuous frames by using iterated function systems, probability functions and window functions.In the last part of this thesis, we introduce the spectral set conjecture. In their investi-gation and study of the conjecture, Jorgensen and Pedersen give a conjecture. Motivated by the conjecture, Lagarias, Reeds and Wang established a characterization of spectra and tilings that can be used to prove the conjecture of Jorgensen and Pedersen by Keller’s criterion. Different techniques to prove these facts have also been developed by Iosevich, Pedersen and Kolountzakis. In this part, we present an elementary approach to give some characterizations of Fourier frames and tilings, and obtain a more general form of spectra and tilings.更多还原