Harmony On The Promotion, With Some Properties Of The Bernoulli Convolution

In this paper, we mainly study the problem about compatible pair and Bernoulli convolutions.Compatible pair is introduced by Strichartz at first. It plays an important role in studying spectral measure. Li,Shen xingcan talk about the relations between the orthogonality and the compatible pair on the prime case. Shen xingcan also investigates the case det(M)|=pa a prime power. This paper weakens the condition of [18] and reprove the corresponding theorem. Also,we talk about the relations between compatible pair and integral self-affine tile. Another part, we study the Bernoulli convolutions. Bernoulli convolutions is a special fractal measure with a parameter. It is a important subject which has been investgated in recent years. We generalize the corresponding results of [21]. The main works are as follows:Firstly, we study the relations between the orthogonality and the compatible pair in the case |det(M)|=pa is a prime power. This paper remove the condition pa-1(Zn)(?)M*(Zn)and improve the theorem of [18]. Furthermore, we study the relations between the compatible pair and integral self-affine tile.Secondly, we talk about the Bernoulli convolutions. We prove a theorem:If for some positive integer k0, p is an odd integer, E(pk0Γ) is a orthogonal basis of L2(μP/2n), then for any positive integer k>0,E(pkΓ) is also a orthogonal basis of L2(μP/2n).