The Orthogonal Spectral Measure

Suppose u is a probability measure with compact support, and set E(A) ={eλ:= e2πiλx,X (?) ∧} for ∧ (?) d, which is a countable set, and we can define Q(ξ)= Σλ(?) |u(ξ+λ)|2 on the Rd.We can get the conclusions:(1)E(A) is an orthogonal set of L2(u), if and only if u(λi - λj) = 0 forλi ≠ λj(?) ∧; (2)E(A) is an orthonormal basis of L2(u), if and only if Q(ξ) = 1 for ξ (?) Rd. Both of them are important theorems in the field of spectrum, which are used in the paper [7] and [8]. They are original from the paper of Jorgensen and Pedersen[14]. But the author did not give a detail proof, we will give a detail proof for them in the paper. Expecially, for the second theorem, i will give two proof methods.The pure style theorem[7]:we can define the u be an F-spectrum measure, then it must be a pure style, is one of the most important theorems in the field of spectrum. For the proof of it, we will make some adjustments according to my own ideas.The problems about the condition of spectrum have been solved in the [7] and [8]. From that, we can know how to construct a spectrum measure, but how to proof a spectrum, we also care about it. So, at the last of the paper, we will give an example to show it.