Spectrality Of The Moran Measures With Two Elements In Digit Sets

Let μ be a Borel probability measure with compact support in Rd.If there exists a countable set Λ 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 correspondingly A is called a spectrum of μ.The problem of the spectrality of measures μ is one of the basic problem for study L2(μ).The graduation thesis is studying the spectrality of Moran measures μ{Pn},{Dn} with {pn}n=1∞ is a sequence of integers bigger than 1 and Dn={0,dn}(?)Z.The main contents of this paper is divided into two chapters:In the third chapter,we study the necessity of spectral measures μ{Pn},{Dn},let{dn}n=1∞ be a bounded odd sequence,if μ{Pn},{Dn} is a spectral measure,then 2|pn for any n>2;let {dn}n=1∞ be a general integer sequence,denote dn=2ind’n,pn=2jnp’n with d’n,p’n are odd.If {d’n}n=1∞ is bounded and μ{Pn},{Dn} is a spectral measure,then the integer sequence {∑k=1n jk-in}n=1∞ are mutually different.In the forth chapter,we study the sufficient conditions of spectral measuresμ{pn},{Dn}.Marks as above,if {∑k=1n jk-in}n=1∞ be a strictly increasing sequence,and 1/2in dn+1/pn+1 for any n>1,then μ{pn},{Dn} is a spectral measure.