【摘要】：本文主要討論了三類數(shù)字集與整數(shù)擴(kuò)張矩陣生成的自仿測度的譜與非譜性質(zhì).首先,利用Strichartz的一個(gè)譜對準(zhǔn)則討論自仿測度的譜性質(zhì),在譜的情形下,找出了它的一些譜.其次,利用自仿測度的Fourier變換零點(diǎn)的分布特點(diǎn)討論了它的非譜性質(zhì),并指出了此時(shí)相互正交的指數(shù)函數(shù)的個(gè)數(shù).本文的內(nèi)容安排如下:第二章討論共線數(shù)字集生成的自仿測度的譜性質(zhì).根據(jù)自仿測度的Fourier變換零點(diǎn)的分布特點(diǎn),來討論整數(shù)擴(kuò)張矩陣與共線數(shù)字集生成的自仿測度的非譜性質(zhì).首先,討論了平面上三元素共線數(shù)字集的非譜性質(zhì).通過求解三個(gè)單位根的和為零的方程,得出自仿測度的Fourier變換零點(diǎn).再利用相似變換下自仿測度的譜性質(zhì)的不變性,從而得出自仿測度的非譜性質(zhì).其次,討論了三角擴(kuò)張矩陣與三元素共線數(shù)字集生成的自仿測度的譜性質(zhì),在是譜的情形下,找到了它的一些譜.最后,討論了q個(gè)元素共線數(shù)字集的情形,利用等比數(shù)列求和公式求得自仿測度的Fourier變換零點(diǎn),得出自仿測度的譜性質(zhì).第三章研究了數(shù)字集有直和分解的情形下生成的自仿測度的譜性質(zhì).譜自仿測度一般由和諧對得到,利用Strichartz的一個(gè)譜對準(zhǔn)則來判定.然而,一些作者已經(jīng)給出了一些不能由和諧對得到譜測度的例子.這里我們給出了更多的不能由和諧對得到譜測度的例子.根據(jù)單位根之和為零的理論知識,若數(shù)字集個(gè)數(shù)大于4時(shí),一般不容易確定其單位根.但在數(shù)字集有直和分解的情形下,我們給出了一些自仿測度的譜性質(zhì).第四章討論零和標(biāo)準(zhǔn)正交基組成的數(shù)字集生成的自仿測度的非譜性質(zhì).具體地,給出了R3中擴(kuò)張矩陣為上三角矩陣產(chǎn)生的自仿測度的非譜性質(zhì).一方面證明了廣義三維Sierpinski墊上自仿測度的譜與非譜性質(zhì).另一方面,給出了整數(shù)擴(kuò)張矩陣是對角矩陣且有兩個(gè)元素相等且為奇數(shù)時(shí)生成的自仿測度的非譜性質(zhì).最后給出了總結(jié),同時(shí)指出進(jìn)一步研究的問題.
[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.
【學(xué)位授予單位】：陜西師范大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O174.12

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本文編號：2230437