The Spectrality Of Discrete Measures And Self-affine Measures

One of the brilliant achievements in the cross-over study of fractal geometry and harmonic analysis is that L2(μ)has an exponential orthogonal basis for the 1/4 Cantor measure μ,therefore,the function in L2(μ)has the Fourier expansion,which is firstly found by Jorgensen and Pederson[19].This amazing-discovery quickly made the Fourier analysis on fractal to be a hot topic of mathematics.The μ is called a spectral measure if there exists an exponential orthogonal basis for L2(μ)(the definition is in chapter 1).In this paper,we consider the spectrality of the self-affine measure μM,D(see(1.1))generated by an expansive matrix M=[a c b d]∈M2(Z),det(M)∈3Z and the three-elements digit set D={(0 0),(α β),(γ η)}(?)Zn,(αη-βγ)(?)3Z.By using mod 3 residue system and the distribution of zeros of the Mask poly-nomial,we divide D and M into 4 classes,{Dj}4j=1 and {Mj}j=14 respectively,where both Dj and Mj contain infinite elements for each j.Fixed Dj and Mj,for any Djk ∈ Dj and Mji ∈ Mj,we prove that μMj,i,Dj,k is a spetral measure.The past literature only studied special digit set with α=η=1,β=γ=0[32].Our research is to consider the class of digit set,and prove that the self-affine measure μMj,i,Dj,k is a spetral measure by classifying the digit set D and expan-sive matrix M.This is the innovation of this paper.Secondly,we study the convolution measure v of discrete measure and Riesz spectral measure in high dimension.We give a sufficient condition in which v is the Riesz spectral measure,and a necessary and sufficient condition in which v is a spectral measure.