广义自相似测度的Fourier变换的几个性质

Some properties on Fourier transform of generalized self-similar measures.

设{Sj}jm=1是Rd上的一族压缩相似映射,Sj(x)=ρjRjx+bj(1≤j≤m),其中0<ρj<1,Rj是d×d维正交矩阵,K是该函数迭代系统的不变集.设{pj}jm=1是Rd上的正连续函数,且{logpj}jm=1满足Dini条件.FANAi-hua等证明了存在惟一的支撑在K上的正则Borel概率测度μ满足λμ=∑mj=1pj(x)μS-j1.本文证明了μ要么关于Lebesgue测度奇异,要么关于Lebesgue测度绝对连续.然后讨论了μ的Fourier变换的渐近性质.

Let Sj （x） =pjRjx＋bj （1≤j≤m） be a family of contractive self-similar mappings on R^d, in which 0〈pj 〈 1 and R： is orthogonal matrix. K is the attractor of the IFS. Let {pj }mj=1 be positive continuous real-valued function on R^d , and {logpi }mj=1 satisfy the Dini condition. There exists the unique probability measure that satisfies the equation λμ = m∑j=1 pj （x）μ · S^-1i . it is proved that if μ is not singular, then it is absolutely continuous with respect to Lebesgue measure. Then the asymptotic behavior of Fourier transform of μ is discussed.