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 $L^p({\mathbb{R}}^d)$ $(1<p<\infty )$ is a $p^{\prime }$-Bessel sequence for some $1<p^{\prime }<\infty$, 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 $p^{\prime }$-Bessel $(C_q)$-system for any $1<p^{\prime }<\infty$. Furthermore, by using techniques from the geometry of Banach spaces, we obtain that, for $1<p\le 2$, no finite disjoint union of translates in $L^p({\mathbb{R}}^d)$ can form an unconditional basis.

12 pages, to appear in Banach J. Math. Appl