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

Bei Liu, Rui Liu

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'…