The Study Of The Spectral Property Of Two Kinds Of Fractal Measures

Let μ be a Borel probability measure with compact support on Rn.We call it a spectral measure if there exists a countable subset Λ(?)Rn such thatεΛ:={e2πi<λ,x>:λ ∈ A} forms an orthonormal basis for L2(μ).In this case,we call A a spectrum of μ and(λ,Λ)a spectral pair,respectively.For an n ×n expanding real matrix M ∈ Mn(R)and a finite digit set DR(?)n with cardinality|D|,let {φd(x)}d∈D be an iterated function system(IFS)defined byφd(x)=M-1(x+d)x ∈Rn,d ∈ D.As an affine iterated function system {φd(x)}d∈D,there exists a unique prob-ability measure μ:=μM,D satisfying the self-affine identity with equal weight:μ=1/|D|Σd∈D μoφd-1,which is supported on the attractor T(M,D)of the IFS{φd}d∈D.In 1998,Jorgensen and Pedersen[43]gave the first singular spec-trum measure.The spectral problem of measure is a basic problem in harmonic analysis.In this paper,we mainly study the spectral problem of two classes of self-affine measures μM,D This paper is divided into four chapters,which specific the arrangement are as follows:In Chapter 1,we first introduce some basic knowledge in this direction,then introduce the research status of fractal measure,and finally list the main conclusions of this paper.In Chapter 2,we study the self-affine measures μM,D generated by a di-agonal matrix M ∈M3(Z)with entries p1,p2,p3∈Z\{0,±1} and a digit set D={(0,0,0)t,(1,0,0)t,(0,1,0)t,(0,0,1)t},and we get:(ⅰ)If two of p1,p2,p3 are odd,then there do not exist infinite families of orthogonal exponential functions in L2(μM,D);(ⅱ)If two of |p1|,|p2|,|p3| are different odd numbers and the other is even,then there exist arbitrary numbers of orthogonal exponential functions in L2(μM,D).This work generalizes the conclusion that Wang[38]published on Math.Nachr which L2(μ)admits 8 mutually orthogonal exponential functions.In Chapter 3,we mainly consider the self-affine measures corresponding to expanding matrix M ∈ M2(Z)and digit set D={(0,0)t,(1,0)t,(2,9)t}.If det(M)∈ 3Z,a necessary and sufficient condition for the existence of an finite orthogonal set of exponential functions in L2(μ)is given.In addition,for a digit set of D={(0,0)t,(1,0)t,(0,1)t},An,He and Tao[49]obtained the necessary and sufficient conditions for it to be a spectral measure if and only if(M,D)is a compatible pair.But for the digit set of we considered,this is not true At the end of the paper we will give a specific example to illustrate that when(M,D)is not compatible,μM,D is still a spectral measureIn Chapter 4,we have summarized the full paper,at the same time,some questions were raised as the goals of our follow-up research.