On complete and incomplete exponential systems

Abstract

Given a bounded domain $\Omega \subset {\mathbb{R}}^d$ with positive measure and a finite set $A=\{a^1,a^2,\dots ,a^d\}$, we say that the set $\mathcal{E}(A)={\{e^{2\pi ix\cdot a^j}\}}_{a^j\in A}$ is a complete exponential system if for every $\xi \in {\mathbb{R}}^d$, there exists $1\le j\le d+1$ 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 $\Omega =B_d$, the unit ball, and when $\Omega =Q_d$, the unit cube. Given a bounded domain $\Omega$, we say that $e^{2\pi ix\cdot a},e^{2\pi ix\cdot a^{\prime }}$ are $\varphi$-approximately orthogonal if $$|{\hat{\chi }}_{\Omega }(a-a^{\prime })|\le \varphi (|a-a^{\prime }|),a\ne a^{\prime }$$ where $\varphi :[0,\infty )\to [0,\infty )$ is a bounded measurable function that tends to $0$ at infinity. We prove that $L^2(B_d)$ does not possess a $\varphi$-approximate orthogonal basis of exponentials for a wide range of functions $\varphi$. 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}).

3 figures

References in corpus (1)

• The Fuglede conjecture for convex domains is true in all dimensions