The Cardinality Of ?_M,D-orthogonal Exponentials Under Non-spectral Self-affine Measures

Let ?M,D be the self-affine measure uniquely determined by the affine iterated function system {?d(x)=M-1(x+d)?d?D,where M ? Mn(R)is a real expanding matrix and D ? Rn is a finite digit set.The non-spectrality of ?M,D is closely connected with the finiteness or infiniteness of orthogonal exponentials in the Hilbert space L2(?M,D).In this thesis,we mainly consider the non-spectral problem of class(?).This kind of problem is to estimate the cardinality of ?M,D-orthogonal exponentials.The main results are the following:(1)Under a class of non-spectral self-affine measures,we obtain a better upper bound on the cardinality of ?M,D-orthogonal exponentials.We provide a more accurate estimate on the cardinality of orthogonal exponentials in the space L2(?M,D)by characterizing the elements of the zero set Z:=Z(mD)?[0,1)n of the symbol function mD(x)in[0,1)n.This extends the results of Dutkay,Jorgensen and others.(2)We consider the non-spectrality of self-affine measure ?M,D corresponding to D={(0,0)t,(?,?)t,h(?,?)t} ? Z2,where ?2+?2?0,h ? Z\{0,1?.We provide some non-spectral conditions for planar self-affine measures with collinear digit set by characterizing the matrix M*k and the zero set Z(mD).Under one of conditions,we prove that there are at most 9 mutually orthogonal exponential functions in L2(?M,D).And under another condition,we construct a class of maximal nine-element orthogonal exponential functions in L2(?M,D).This enriches the research on the spectrality and non-spectrality of self-affine measures with collinear digit set.