A Spectral Eigenvalue Problem Of Spectral Self-similar Measures

Let μ be a Borel probability measure with compact support in i0·1If there is a complex exponential function family E(∧):= {e-2πi<λ,x):λ∈A}constitutes an orthonormal basis in L2(μ),we call μ a spectral measure.In this case,A is called a spectrum of μ.Let q>1 and let D C Z be a finite set.Then by Hutchinson’s theorem,the self-similar measure with equal weights generated by the pair(q,D)is the unique Borel probability measure which satisfies the invariance equationμq.D(E)=1/#D(?)μq,D(qE-d),for each Borel set E.This dissertation is divided into two parts.We consider the self-similar measure μq.{0,ar,br},where r=q/3 in the first part.Fu yansong et al.[53]characterized when it is a spectral mea-sure completely.We mainly consider the spectral structure problem of the spectral measure.We study the spectral structure of the spectral measure,obtain a simple tree structure represen-tation,and obtain a sufficient condition from the largest orthogonal family to the orthogonal basis.The condition contains almost all known sufficient conditions(need appropriate ad-justments).As an application,we settle the spectral eigenvalue problem of the above spectral measure with respect to the model spectrum A completely,i.e.,find all real numbers t such that t∧ is also a spectrum of μq.{0,ar,br} It is worth to point out that our results is true forμ4.{0,2} or a more general self-similar spectral measure by appropriate adjustments.The re-sults of this section and the work published by the author and collaborators in J.Funct.Anal.are the basis for the author’s follow-up study.We then consider a class of self-similar measures μq,(0,r,...,(b-1)r},r=q/b∈Z with consecutive sets.It is known that the set∧w=(?){0,1,...,q-1}wkpk-1 is a spectrum of μq,{0.r,...,(b-1)r} for any w=w1w2…∈{-1,1}∞.The ∧w is called a model random spectrum of μq,{0,r,...,(b-1)r}.The common spectral eigenvalue problem with respect to the model random spectrum is proposed in[51].We answer the above open question partially and give a class of common spectral eigenvalues.