Spectral Properties Of Certain Moran Measures

Fractal geometry is a new subject,which was proposed by Mandelbrot in 1975.It not only combines with other branches of mathematics(such as probability theory,number theory,harmonic analysis,complex analysis etc.),but also promotes the development of other disciplines(such as physics,chemistry,biology,geography,engineering etc.),and it is widely concerned by the scientific community.In 1998,Jorgensen and Pedersen discovered that the function on the space of L2(?)corresponding to the 1/4Cantor measure(the first singular non atomic spectral measure)had Fourier expansion,which made the Fourier analysis on the fractal set a hot topic in mathematical research.In 2000,R.Strichartz extended the spectral study of self-affine measure to Moran measure,which attracted the attention of a group of mathematicians.This thesis mainly studies the spectral properties of Moran measure.The previous research on the Moran measure was limited to certain special cases,and this thesis expand it to more extensive cases.Let Rk,Dk be expanding matrix and digit set,respectively,and ak,bk,qk,pk?N\{1}.In this thesis,we mainly study the spectral properties of the Moran measure ?{Rk}{Dk},which is generated by the following kinds of Rk and Dk:(?)Rk=diag(ak,bk),Dk={0,1,…,qk-1}v,with v=(?1,?2)t?N2;(?)Rk=J(pk)or(?),Dk={0,1,…,qk-1?v,with v=(1,1)t;(?)Rk=diag{a1,k,a2,k,…an,k},ai,k?N\{1}(i=1,2,…,n),Dk={0,1,…,qk-1}e1+…+{0,1,…,qk-1}en,with ei is unit vector in ndimensional in which the i-th element is 1 and the remaining elements are 0.This thesis firstly proves that the sufficient and necessary conditions for the existence of the compatible pair in a class of situation.Secondly,we use the compatible pair to construct the orthogonal set of ?{Rk}{Dk},and finally prove the completeness of the orthogonal set,so as to complete the proof of spectral property.