非谱自仿测度下正交指数函数系的基数

The Cardinality of Orthogonal Exponentials Under the Non-spectral Self-affine Measures

设μ_(M,D)是由扩张矩阵M∈M_n(Z)和有限数字集D?Z^n通过仿射迭代函数系统{φ_d(x)=M^(-1)(x+d)}_(d∈D)唯一确定的自仿测度,它的非谱性与相应的平方可积函数构成的Hilbert空间L^2(μ_(M,D))中正交指数函数系的有限性或无限性密切相关.通过对数字集D的符号函数m_D(x)的零点集合Z(m_D)的特征分析以及其中非零中间点(即坐标为0或1/2的点)和非中间点的性质应用,得到了非谱自仿测度下正交指数函数系基数的一个更为精确的估计,改进推广了Dutkay,Jorgensen等人的相关结果.

Let μM,D be the self-affine measure uniquely determined by an expanding matrix M E Mn（Z） and a finite digit set D C Zn through the aitine iterated function system （IFS）{Фd（x） ---- M^-1（x ＋ d）}deD. The non-spectrality of #M,D is directly con- nected with the finiteness or infiniteness of orthogonal exponentials in the Hilbert space L2（ItM,D）. We provide a better estimate on the cardinality of μM,D-Orthogonal expo- nentials by characterizing the zero set Z（mD） of the symbol function mD（x） and its middle points. The results here extend the corresponding results of Dutkay, Jorgensen and others.