Spectral Properties Of Two Kinds Of Moran Measures On The Line

Let ? be a Borel probability measure with compact support on Rn,if there exists a countable set A???Rⁿ such that E? =?e^2?i<?,x>:???} forms an orthogonal basis of L²???,then ? is called a spectral measure and ? is called a spectrum for?.Fractal spectral measure is a basic problem in the study of Fourier analysis on Fractal,it originated from the spectral set conjecture proposed by Fuglede in 1974 and the first fractal spectral measure given by Jorgensen and Pedersen in 1998.In this thesis,we will study the spectral problems of two kinds of Moran measures on the line.This thesis is divided into three chapters and arranged as follows:In the first chapter,we make a brief summary of research background and research status of fractal spectral measure,at the same time,we introduce some preparatory knowledge and the main results of this study.In the second chapter,we mainly consider the spectrality of a class of Moran measure generated by direct sum digit sets on the line.Let ?= N^s+1,D_n ={0,1,…,N-1}???N^t_nl_n{0,1,…,N-1},where s,N>2 are positive integers,and {t_n}_n=1^?,{l_n}_n=1^? are two bounded positive integer sequences with ln?Z\NZ.We obtain a sufficient condition for the Moran measure ?_?,{D_n} generated by p and {D_n}_n=1^? to be a spectral measure.In the third chapter,we mainly study the spectral problems of a class of Moran measure generated by four-element digit sets on the line.Let ?_<sup>s_n,D_n ={0,a_n,2^t L_n,a_n+2^tL_n'},where {s_n}_n=1^? is a sequence of integers greater than 1,{a_n,L_n,L_n'}_n=1^? is a sequence of bounded odd numbers,and t is a positive integer.We obtain some sufficient conditions for the Moran measure ?{?_n},{D_n} to be a spectral measure.