Spectral Property Of Self-similar Measures With Consecutive Digits

Let ?be a compactly supported Borel probability measure on Rd.We say that?is a spectral measure if there exists a countable set A(?)Rd so that E(A):?{e2?i<?,x>:??A} is an orthonormal basis for L2(?).In this case,A is called a spectrum of ? In this thesis,we will mainly discuss about the spectral property of the self-similar measure ?b,D,which is generated by the common contraction ratio 1/b and the digits D={0,r,2r,…,r(g-1)} with b=qr.The main work of this thesis is spread in the following three chapters:In Chapter two,We will introduce some basic knowledge and tools for the re-search of spectral measures.It mainly introduces the basic properties of the iterated function system,the simple nature of the spectral measure and the nature of its maximal orthogonal set.In Chapter three,We will give a sufficient condition for ?b,D to be a spectral measure.We characterize all the maximal orthogonal sets A when q divides b via a maximal mapping on the q-adic tree.Then We give a sufficient condition for ?b,D to be a spectral measure.In Chapter four,We will give a sufficient condition that ?b,D is not a spectral measure.We find that the spectral property of ?b,D is related to the number of non-zero elements of the b-adic expansion of elements in A.