On complete and incomplete exponential systems

TL;DR

This paper classifies complete and incomplete exponential systems on unit balls and cubes, and shows that certain approximate orthogonal bases of exponentials do not exist in these domains, connecting to distance set theory.

Contribution

It provides a classification of exponential systems on specific domains and proves non-existence of approximate orthogonal bases in these settings.

Findings

Complete systems characterized on unit ball and cube

Incomplete systems identified under certain conditions

No $$-approximate orthogonal basis of exponentials in $L^2(B_d)$

Abstract

Given a bounded domain $\Omega \subset {\Bbb R}^d$ with positive measure and a finite set $A=\{a^1, a^2, \dots, a^d\}$, we say that the set ${\mathcal E}(A)={\{e^{2 \pi i x \cdot a^j}\}}_{a^j \in A}$ is a complete exponential system if for every $\xi \in {\Bbb R}^d$, there exists $1 \leq j \leq d+1$ such that \begin{equation} \label{completedef} \int_{\Omega} e^{-2 \pi i x \cdot (a^j-\xi)} 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 $e^{2 \pi i x \cdot a}, e^{2 \pi i x \cdot a'}$ are $\phi$-approximately orthogonal if $$|\widehat{\chi}_{\Omega}(a-a')| \leq \phi(|a-a'|), \ a\neq a'$$ where $\phi: [0, \infty) \to [0, \infty)$ is a bounded measurable function that tends to $0$ at infinity. We prove that $L^2(B_d)$ does not possess a $\phi$-approximate orthogonal basis of exponentials for a wide range of functions $\phi$. 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}).