Upper Beurling Density of Systems formed by Translates of finite Sets of Elements in $L^p(\R^d)$

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 $L^p(\R^d)$ $(1<p<\infty)$ is a $p'$-Bessel sequence for some $1<p'<\infty$, 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 $L^p(\R^d)$ can form a $p'$-Bessel $(C_q)$-system for any $1< p'<\infty$. Furthermore, by using techniques from the geometry of Banach spaces, we obtain that, for $1<p\le2$, no finite disjoint union of translates in $L^p(\R^d)$ can form an unconditional basis.