The Conditions Of Infinite Orthogonal Exponentials And The Spectrality Of Certain Spatial Self-affine Measures

The question of determining the spectrality or non-spectrality of a self-affine measures μM,D is one of the important subject in the spectral theory of self-affine measures. In this regard, the finiteness or infiniteness of μM,D-orthogonal exponen-tials plays a central role. So, this paper mainly analysies the conditions of infinite orthogonal exponentials and the spectrality of certain spatial self-affine measures. We have the following results:In the first part, by using the middle points of the zero set Z(mD) of the function mD(x), we obtain the condition of infinite orthogonal exponentials on the spatial self-affine measures. Such research is necessary for further understanding the spectrality of self-affine measures. And, we have some applications on this result.In the second part, we are mainly to analyse the spectrality of the affine-measures μM,D produced from iterated function system {φd(x)]d∈D when M= 1/2[p1+p2, p1-p3,p2-p3; p1-p2, p1+p3;-P2+P3; -p1+ p2, -p1+p3; p2+p3], D= {0,e1,e2,e3} are in the space R3，where pj ∈ Z \{0,±1}(j= 1,2,3)，ex, e2, e3 are the standard basis of unit column vectors in R3，and obtain that (1)if pj∈2Z\{0,2}(j=1,2,3) or p1=p2=p3 =2，then μMD is a spectral measure; (2)if there is at least one even number among p1,p2,p3, then there are infinite families of orthogonal exponentials E(A) in L2(μMD) and ∧∈Z3; (3)if pj∈2Z+1\{±l}(j=1,2,3), then μM,D is a non-spectral measure, and there exist at most 4 mutuaUy orthogonal exponential functions in L2(μM,D), where the number 4 is the best.