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

DOI：10.16205/j.cnki.cama.2019.0035

 作者 单位 E-mail 李娜 陕西师范大学数学与信息科学学院, 陕西　西安 710119. jllimath10@snnu.edu.com 李建林 陕西师范大学数学与信息科学学院, 陕西　西安 710119. jllimath10@snnu.edu.com

设$\mu_{M,D}$是由仿射迭代函数系$\{\phi_{d}(x)=M^{-1}(x+d)\}_{d\in D}$唯一确定的自仿测度, 它的谱与非谱性质与Hilbert空间$L^{2}(\mu_{M,D})$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$\mu_{M,D}$\,{-}\!\!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$\mu_{M,D}$的非谱性的判断方面便于直接验证.

Let $\mu_{M,D}$ be a self-affine measure uniquely determined by the iterated function system $\{\phi_{d}(x)=M^{-1}(x+d)\}_{d\in D}$.\ The spectrality or non-spectrality of $\mu_{M,D}$ is directly connected with the finiteness or infiniteness of orthogonal exponentials in the Hilbert space $L^{2}(\mu_{M,D})$. In this paper, the authors provide a sufficient condition for the finite $\mu_{M,D}${-}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 $\mu_{M,D}$.