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

Abstract:

Let $1\le p <\infty$, $f\in L_p(\mathbb R)$, and $\Lambda \subseteq \mathbb R$. We consider the closed subspace of $L_p(\mathbb R)$, $X_p (f,\Lambda )$, 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 $L_1(\mathbb R)$, 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 $L_p(\mathbb R)$, 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 $f\in L_p(\mathbb R)$ and $\Lambda \subseteq \mathbb Z$ so that $\{f_{(\lambda )} :\lambda \in \Lambda \}$ is unconditional basic and $L_p(\mathbb R)$ embeds isomorphically into $X_p (f,\Lambda )$.