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

TL;DR

This paper investigates the density properties of systems formed by translates of finite sets in $L^p( ^d)$, establishing conditions under which such systems can or cannot be Bessel sequences, $C_q$-systems, or unconditional bases.

Contribution

It proves that finite disjoint unions of translates in $L^p( ^d)$ cannot form certain structured systems like Bessel sequences or unconditional bases, depending on their density and geometric properties.

Findings

Finite disjoint unions of translates with Bessel property have finite upper Beurling density.

Such unions cannot be $C_q$-systems due to infinite density requirements.

For $1<p extless 2$, these unions cannot form unconditional bases.

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.