Spectrality Of The Moran Measures With Two Elements In Digit Sets

Let ? be a Borel probability measure with compact support in Rd.If there exists a countable set ? such that E(?):={e2?i<?,x>:???}forms an orthonormal basis for L2(?),the measure ? is called a spectral measure and correspondingly A is called a spectrum of ?.The problem of the spectrality of measures ? is one of the basic problem for study L2(?).The graduation thesis is studying the spectrality of Moran measures ?{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 ?{Pn},{Dn},let{dn}n=1? be a bounded odd sequence,if ?{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 ?{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 measures?{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 ?{pn},{Dn} is a spectral measure.