Self-similar measures and their Fourier transforms. II
Self-similar measures and their Fourier transforms. II
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DOI：
10.1090/s0002-9947-1993-1081941-2
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发表时间：
1993
影响因子：
1.3
通讯作者：
R. Strichartz
中科院分区：
文献类型：
--
作者：
R. Strichartz
Hutchinson将Rn上的自相似测度定义为满足(公式…)的概率测度。这里S jx=ρj Rj x+bj是压缩相似(0<；ρj<；1，Rj正交)，权aj满足0<；Aj<；1，∑j=1maj=1。类推，我们用相同的恒等式定义了自相似分布*)，但允许权aj是任意复数。给出了(*)在紧支集分布中存在解的充要条件，并证明了此类解的空间总是有限维的。如果F表示紧支集的自相似分布的傅里叶变换，令(公式…)其中β由公式∑j=1 mρj−β|a j|2=1定义
A self-similar measure on R n was defined by Hutchinson to be a probability measure satisfying (formule...) here S j x = ρ j R j x+b j is a contractive similarity (0 < ρ j < 1, R j orthogonal) and the weights a j satisfy 0 < A j < 1, ∑ j=1 m a j = 1. By analogy, we define a self-similar distribution by the same identity *) but allowing the weights a j to be arbitrary complex numbers. We give necessary and sufficient conditions for the existence of a solution to (*) among distributions of compact support, and show that the space of such solutions is always finite dimensional. If F denotes the Fourier transformation of a self-similar distribution of compact support, let (formule...) where β is defined by the equation ∑ j=1 m ρ j −β |a j | 2 = 1