# Systems formed by translates of one element in $L pmathbb R$ - Mathematics > Functional Analysis

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Abstract: Let $1\le p <\infty$, $f\in L p eal$ and $\Lambda\subseteq eal$. Weconsider the closed subspace of $L p eal$, $X p f,\Lambda$, generated bythe 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$L 1 eal$, we prove that $X 1 f,\Lambda$ embeds almost isometrically into$\ell 1$. If $\{f {\lambda} :\lambda\in\Lambda\}$ is an unconditional basicsequence in $L p eal$, then $\{f {\lambda} : \lambda\in\Lambda\}$ isequivalent to the unit vector basis of $\ell p$ for $1\le p\le 2$ and $X pf,\Lambda$ embeds into $\ell p$ if $2

4$, there exists $f\inL p eal$ and $\Lambda \subseteq \zed$ so that $\{f {\lambda}:\lambda\in\Lambda\}$ is unconditional basic and $L p eal$ embedsisomorphically into $X p f,\Lambda$.

Author: ** E. Odell, B. Sari, Th. Schlumprecht, B. Zheng**

Source: https://arxiv.org/