Spectrum Property Of A Class Of Moran Measures

Fractal geometry is a popular research subject,there are many cross stud-ies on fractal,and Fourier analysis on Fractal has become a hot research topic in recent years.The existence of orthogonal basis of exponential functions,or the spectral measure problem is one of the basic problems of Fourier analysis on Fractal.R.Strichartz introduced the spectral measure problem to Moran mea-sures and proved some sufficient conditions for a class of Moran measures to be spectral measures.Recently An Li-xiang and He Xing-gang proved that spectral properties of Moran measures in one dimension.We consider spectral proper-ties of Moran measures in two dimension.let(?)be an expanding positive integer matrix,and let Dk = 0,1…,qk-1}v1 + {0,1,…,qk-1}v2,where v1 =(1,0)t,v2 =(0,1)t and qk>1 is a integer number,this thesis main-ly studies the existence of the Moran measure μ{Rk}{Dk},which is generated by{Rk}∞k=1 and {Dk}∞k=1 as followingμ{Rk} {Dk}:＝ δR1-1D1*δ(R2R1)-1D2*…*δ(Rk…R2R1)-1Dk*….We prove that μ{Rk){Dk} is a spectral measure if qk|ak and qk|bk.This thesis consists of five chapters,the specific arrangement is as follows:In the first chapter,we give a brief summary of research background,re-search meaning and recent developments of spectral measures.Besides,we introduce some main results of this thesis.In the second chapter,we introduce some basic concepts of spectral mea-sures and construct some lemmas we need in the proofs.In the third chapter,we discuss the convergence of Fourier transform of infinity convolutions,we prove the existence of the Moran measure μ{Rk}{Dk}and a property of admission pairs.In the fourth chapter,we prove the spectrum of Moran measure μ{Rk}{Dk}through studying the expanding matrix Rk,it also gives the construction of the spectra.In the fifth chapter,we give some examples and corollaries with different Dk.