有限μM,D-正交指数函数系的一个充分条件

A Sufficient Condition for the Finite μM,D-Orthogonal Exponentials Function System

设μM,D是由仿射迭代函数系{φd(x)=M^-1-(x+d)}d∈D唯一确定的自仿测度,它的谱与非谱性质与Hilbert空间L^2(μM,D)中正交指数函数系的有限性和无限性有着直接的关系.本文将利用矩阵的初等变换给出μM,D正交指数函数系有限性的一个充分条件.由于这个条件只与矩阵M的行列式有关,因此,它在μM,D的非谱性的判断方面便于直接验证.

Let μM,D be a self-affine measure uniquely determined by the iterated function system {φd(x)=M^-1(x+d)}d∈D.The spectrality or non-spectrality of μM,D is directly connected with the finiteness or infiniteness of orthogonal exponentials in the Hilbert space L^2(μM,D).In this paper,the authors provide a sufficient condition for the finite μM,D-orthogonal exponentials by applying the elementary matrix transformations.This sufficient condition depends only upon the determinant of the matrix M,and is easy to use in the research of non-spectrality of μM,D.