Systems formed by translates of one element in $L_p(\mathbb{R})$

Abstract

Let $1\le p<\infty$, $f\in L_p(\Re )$ and $\Lambda \subseteq \Re$. We consider the closed subspace of $L_p(\Re )$, $X_p(f,\Lambda )$, generated by the set of translations $f_{(\lambda )}$ of $f$ by $\lambda \in \Lambda$. If $p=1$ and $\{f_{(\lambda )}:\lambda \in \Lambda \}$ is a bounded minimal system in $L_1(\Re )$, we prove that $X_1(f,\Lambda )$ embeds almost isometrically into ${\ell }_1$. If $\{f_{(\lambda )}:\lambda \in \Lambda \}$ is an unconditional basic sequence in $L_p(\Re )$, then $\{f_{(\lambda )}:\lambda \in \Lambda \}$ is equivalent to the unit vector basis of ${\ell }_p$ for $1\le p\le 2$ and $X_p(f,\Lambda )$ embeds into ${\ell }_p$ if $2<p\le 4$. If $p>4$, there exists $f\in L_p(\Re )$ and $Λ\subseteq \zed$ so that $\{f_{(\lambda )}:\lambda \in \Lambda \}$ is unconditional basic and $L_p(\Re )$ embeds isomorphically into $X_p(f,\Lambda )$.