一类特定数字集下自仿测度的非谱性

Non-Spectrality of Self-Affine Measures on A Class of Specific Set of Digits

主要讨论与扩张矩阵M=diag[p_1,p_2,p_3](p_j∈Z\{0,±1},j=1,2,3)和数字集D={0,e_1,e_2,e_3,e_1+e_2,e_1+e_3,e_2+e_3,e_1+e_2+e_3}所对应的自仿测度μ_(M,D)的谱性,这里e_1,e_2,e_3是空间R^3中的标准正交基.通过分析Fourier变换μ_(M,D)的零点Z(μ_(M,D))的特征,证明当p_j∈2Z+1\{0,±1},j=1,2,3,μ_(M,D)是非谱测度,空间L^2(μ_(M,D))中正交指数函数系至多包含"8"个,且数字"8"是最好的,推广了文献[14]相关的结果.

Abstract： In this paper, the spectral properties of the self-affine measure/XM,O corresponding to the expanding matrix M = diagM=diag[p_1,p_2,p_3]（p_j∈Z／{0,±1},j=1,2,3）and the digital set D ={0,e_1,e_2,e_3,e_1＋e_2,e_1＋e_3,e_2＋e_3,e_1＋e_2＋e_3}are discussed. Here e1 ,e： ,e3 is a standard or-thonormal base in space. By analyzing the characteristics of the zero point Z（μ_（M,D）of the Fou-rier transform μM,D,it is proved that when p_j∈2Z＋1／{0,±1},j=1,2,3,μ_（M,D）contains at most ＂8＂, and the number ＂8＂ is the best, which extends the results of the literature [ 14].