The Study Of Spectrality And Spectral Structure Of Two Kinds Of Singular Measures

Let ? be a Borel probability measure with compact support on Rd.We call ? a spectral measure if there exists a discrete set A(?)Rd such that the exponential function family E(?):={e2?i<?x>:???} forms an orthonormal basis for L2(?).And we call A a spectrum for ?.The spectral measure supported on the fractal set is called the fractal spectral measure,which is the natural extension of Fourier analysis in the field of Fractal geometry.Compared with the Lebesgue spectral measure,the known fractal spectral measure contains uncountably many spectra with 0,so the spectrum of the fractal spectral measure has a complicated structure.In this dissertation,we mainly study the conditions for two kinds of fractal measures becoming spectral measures and the spectral structure of a class of fractal spectral measure.This dissertation is divided into five chapters:In Chapter 1,we introduce the background of the research,motivation and main results;In Chapter 2,we give the basic knowledge and related tools needed for the dissertation.The core parts of this dissertation are from Chapter 3 to Chapter 5,briefly introduced as follows:In Chapter 3,we study the self-similar measure ??,D generated by D={(0,0)T,(1,0)T,(0,1)T,(-1,-1)T} and matrix ?-1 I.We completely describe the sufficient and necessary conditions for this measure becoming a spectral one.The result is published on Fractals-complex geometry patterns and scaling in nature and society.In Chapter 4,we study the spectral properties of a Moran measure ?{Rk},{Dk},where Rk is the entire expansion matrix,Dk={0,1,…,qk-1}v is a digit set and qk?2,v?Zd\{0}.Under the condition of all(Rk,Dk)being admissible,we obtain a sufficient condition for the measure ?(Rk},{Dk} being a spectral one.The result has been submitted to the Journal of Mathematical Analysis and Applications.In Chapter 5,we study the spectral structure of a class of random convolution spectral measure.First,we obtain three sufficient and necessary conditions for the or-thonormal set ?(q,L)being a spectrum.Then we apply it to the ternary digit type of random convolution spectral measure,and solve the first type of spectral eigenval-ue problem of this measure which has common symmetric spectrum.The results are published on Analysis mathematica.