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

Let μM,D be the self-affine measure uniquely determined by the affine iterated function system {φd（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 μM,D is closely connected with the finiteness or infiniteness of orthogonal exponentials in the Hilbert space L2（μM,D）.In this thesis,we mainly consider the non-spectral problem of class（Ⅰ）.This kind of problem is to estimate the cardinality of μM,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 μM,D-orthogonal exponentials.We provide a more accurate estimate on the cardinality of orthogonal exponentials in the space L2（μM,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 μM,D corresponding to D={（0,0）t,（α,β）t,h（α,β）t} ? Z2,where α2+β2≠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（μM,D）.And under another condition,we construct a class of maximal nine-element orthogonal exponential functions in L2（μM,D）.This enriches the research on the spectrality and non-spectrality of self-affine measures with collinear digit set.