Three Methods Of Proving Spectral Property Of Self-similar Measure By Using Compatible Pair

Let μ be a Borel probability measure with compact support in Rd.If there exists a countable set Λ(?)Rd such that E(Λ):={e2πi<λ,x>:λ∈Λ} forms an orthonormal basis for L2(μ),the measure μ is called a spectral measure and Λ is called a spectrum of μ.This paper summarizes the papers of Strichartz,Laba and Wang Yang and Dutkay,Haussermann and Lai Chun-kit.The main contribution of this paper is to systematically organize,modify and simplify the relevant theories and their proofs,so as to prepare for the follow-up research.The main content of this paper is divided into three sections.In section three,we introduce Strichartz’s paper.Under the condition of Z(δD)∩T(b,C)＝(?),they prove that the self-similar measure generated by the compatible pair is spectral measure by using finite convolution approximation.In section four,we introduce the paper of Laba and Wang Yang,Laba and Wang Yang improved on the Strichartz paper.They removed the condition of Z(δD)∩T(b,C)=(?)and proved the spectral property of self-similar measure by using the Ruelle operator.In section five,we introduce Dutkay,Haussermann and Lai Chun-kit’s paper,they used the periodic zero set of Fourier transform Z(μb,D):={ξ∈R:μb,D(ξ+k)=0,(?)k ∈ Z} to prove the spectral property.Meanwhile,this paper makes further improvement on the proof.