论文标题
完全且不完整的指数系统
On complete and incomplete exponential systems
论文作者
论文摘要
给定一个有限的域$ω\ subset {\ bbb r}^d $，并带有积极的措施，有限集$ a = \ {a^1，a^2，\ dots，a^d \} $，我们说集合$ {\ Mathcal e}（\ Mathcal e}（a）= {a）= { A^J}}} _ {a} $中的a^j \是一个完整的指数系统，如果对于{\ bbb r}^d $中的每个$ξ\，存在$ 1 \ leq j \ leq j \ leq d+1 $ （a^j-ξ）} dx \ not = 0; \ end {equation}否则$ {\ mathcal e}（a）$称为不完整的指数系统。在本文中，当$ω= b_d $，单位球以及$ω= q_d $，单位立方体时，我们本质上对完整和不完整的指数系统进行了分类。 给定一个有限的域$ω$，我们说$ e^{2πix \ cdot a}，e^{2πix \ cdot a'} $是$ ϕ $ - approx，如果$ | \ leq ϕ（| a-a'|），\ a \ a \ neq a'$$其中$ ϕ：[0，\ infty）\ to [0，\ infty）$是一个有界的可测量函数，在infinity时趋向于$ 0 $。我们证明，$ l^2（b_d）$不具备广泛函数$ ϕ $的指数的$ ϕ $ -Approximate正交基础。证明涉及与Furstenberg，Katznelson和Weiss（\ cite {fkw90}）开发的阳性Lebesgue上部密度集中距离理论的联系。
Given a bounded domain $Ω\subset {\Bbb R}^d$ with positive measure and a finite set $A=\{a^1, a^2, \dots, a^d\}$, we say that the set ${\mathcal E}(A)={\{e^{2 πi x \cdot a^j}\}}_{a^j \in A}$ is a complete exponential system if for every $ξ\in {\Bbb R}^d$, there exists $1 \leq j \leq d+1$ such that \begin{equation} \label{completedef} \int_Ω e^{-2 πi x \cdot (a^j-ξ)} dx \not=0; \end{equation} otherwise ${\mathcal E}(A)$ is called an incomplete exponential system. In this paper, we essentially classify complete and incomplete exponential systems when $Ω=B_d$, the unit ball, and when $Ω=Q_d$, the unit cube. Given a bounded domain $Ω$, we say that $e^{2 πi x \cdot a}, e^{2 πi x \cdot a'}$ are $ϕ$-approximately orthogonal if $$|\widehatχ_Ω(a-a')| \leq ϕ(|a-a'|), \ a\neq a'$$ where $ϕ: [0, \infty) \to [0, \infty)$ is a bounded measurable function that tends to $0$ at infinity. We prove that $L^2(B_d)$ does not possess a $ϕ$-approximate orthogonal basis of exponentials for a wide range of functions $ϕ$. The proof involves connections with the theory of distances in sets of positive Lebesgue upper density originally developed by Furstenberg, Katznelson and Weiss (\cite{FKW90}).
