关于Bernoulli卷积的一个谱性质

A spectral property of the Bernoulli convolutions

设μρ是R上的参数为0<ρ<1的Bernoulli卷积测度.Hu和Lau证明了L2(μ)空间中含有一个无限指数型正交序列的充要条件是ρ是分数p/q的方根,其中p为奇数,q为偶数.本文讨论当ρ为其余情形时,指数型正交序列集中元素个数,证明了当ρ-1不是奇数型方程的根或者其本原最小多项式g(x)满足g(l)为奇数时,则L2(μ)中任何指数型正交集至多含有两个元素.

Let be the Bernoulli convolution associated with on . Hu and Lau proved that the space contained an infinite orthonormal set of exponential functions if and only if is the nth root of a fraction where is odd and even. In this paper, we discuss the number of the orthonormal set of exponential function in other cases. And we prove that either isn＇t a root of a equation of ＂odd＂ type or is even, any orthonormal set of exponential function in has at most two elements, where is the minimal primitives polynomial for.