Upper Beurling density of systems formed by translates of finite Sets of elements in $L^p(\mathbb{R}^d)$

Abstract

In this paper, we prove that if a finite disjoint union of translates $\bigcup_{k=1}^n\{f_k(x-\gamma)\}_{\gamma\in\Gamma_k}$ in $L^p(\mathbb{R}^d)$ $(p \in (1, \infty))$ is a $p'$-Bessel sequence for some $p' \in (1, \infty)$, then the disjoint union $\Gamma=\bigcup_{k=1}^n \Gamma_k$ has finite upper Beurling density, and that if $\bigcup_{k=1}^n\{f_k(x-\gamma)\}_{\gamma\in\Gamma_k}$ is a $(C_q)$-system with $1/p+1/q=1$, then $\Gamma$ has infinite upper Beurling density. Thus, no finite disjoint union of translates in $L^p(\mathbb{R}^d)$ can form a $p'$-Bessel $(C_q)$-system for any $p'\in (1,\infty)$. Furthermore, by using techniques from the geometry of Banach spaces, we obtain that, for $p \in (1, \le2)$, no finite disjoint union of translates in $L^p(\mathbb{R}^d)$ can form an unconditional basis.