三类数字集产生的自仿测度的谱性

发布时间：2018-09-08 11:52

【摘要】：本文主要讨论了三类数字集与整数扩张矩阵生成的自仿测度的谱与非谱性质.首先,利用Strichartz的一个谱对准则讨论自仿测度的谱性质,在谱的情形下,找出了它的一些谱.其次,利用自仿测度的Fourier变换零点的分布特点讨论了它的非谱性质,并指出了此时相互正交的指数函数的个数.本文的内容安排如下:第二章讨论共线数字集生成的自仿测度的谱性质.根据自仿测度的Fourier变换零点的分布特点,来讨论整数扩张矩阵与共线数字集生成的自仿测度的非谱性质.首先,讨论了平面上三元素共线数字集的非谱性质.通过求解三个单位根的和为零的方程,得出自仿测度的Fourier变换零点.再利用相似变换下自仿测度的谱性质的不变性,从而得出自仿测度的非谱性质.其次,讨论了三角扩张矩阵与三元素共线数字集生成的自仿测度的谱性质,在是谱的情形下,找到了它的一些谱.最后,讨论了q个元素共线数字集的情形,利用等比数列求和公式求得自仿测度的Fourier变换零点,得出自仿测度的谱性质.第三章研究了数字集有直和分解的情形下生成的自仿测度的谱性质.谱自仿测度一般由和谐对得到,利用Strichartz的一个谱对准则来判定.然而,一些作者已经给出了一些不能由和谐对得到谱测度的例子.这里我们给出了更多的不能由和谐对得到谱测度的例子.根据单位根之和为零的理论知识,若数字集个数大于4时,一般不容易确定其单位根.但在数字集有直和分解的情形下,我们给出了一些自仿测度的谱性质.第四章讨论零和标准正交基组成的数字集生成的自仿测度的非谱性质.具体地,给出了R3中扩张矩阵为上三角矩阵产生的自仿测度的非谱性质.一方面证明了广义三维Sierpinski垫上自仿测度的谱与非谱性质.另一方面,给出了整数扩张矩阵是对角矩阵且有两个元素相等且为奇数时生成的自仿测度的非谱性质.最后给出了总结,同时指出进一步研究的问题.
[Abstract]:In this paper, we mainly discuss the spectral and non-spectral properties of self-affine measures generated by three types of digital sets and integer expansion matrices. Firstly, we discuss the spectral properties of self-affine measures by using a spectral pairing principle of Strichartz. In the case of spectrum, we find out some spectra of self-affine measures. Secondly, we discuss its non-affine measures by using the distribution characteristics of zero points of Fourier transform of self-affine measures. In the second chapter, we discuss the spectral properties of the self-affine measures generated by collinear digital sets. According to the distribution characteristics of the zeros of the Fourier transform of the self-affine measures, we discuss the non-spectral properties of the self-affine measures generated by the integer expansion matrix and the collinear digital sets. Firstly, the non-spectral properties of the three-element collinear digital set on the plane are discussed. The Fourier transform zeros of the self-affine measure are obtained by solving the equation that the sum of three unit roots is zero. Then the non-spectral properties of the self-affine measure are obtained by using the invariance of the spectral properties of the self-affine measure under the similar transformation. Secondly, the triangular expansion matrix and the three elements are discussed. The spectral properties of the self-affine measure generated by a collinear digital set are found under the condition that it is a spectrum. Finally, the case of a q-element collinear digital set is discussed. The Fourier transform zeros of the self-affine measure are obtained by using the summation formula of an equal ratio sequence, and the spectral properties of the self-affine measure are obtained. The spectral properties of the self-affine measure generated by a pair of harmonic pairs are generally determined by a spectral alignment criterion of Strichartz. However, some authors have given some examples of spectral measures which can not be obtained by a pair of harmonic pairs. Here we give more examples of spectral measures which can not be obtained by a pair of harmonic pairs. In general, it is not easy to determine the unit root if the number of digital sets is more than 4. But in the case of direct sum decomposition of digital sets, we give some spectral properties of self-affine measures. On the one hand, the spectral and spectral properties of the self-affine measure on the generalized three-dimensional Sierpinski mat are proved. On the other hand, the non-spectral properties of the self-affine measure generated on the generalized three-dimensional Sierpinski mat are given when the integer expansion matrix is a diagonal matrix with two equal elements and is an odd number. At the same time, further research is pointed out.
【学位授予单位】：陕西师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O174.12

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