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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: E. Odell, B. Sarı, Th. Schlumprecht and B. Zheng
Journal: Trans. Amer. Math. Soc. 363 (2011), 6505-6529
MSC (2010): Primary 42C30, 46E30; Secondary 46B15
DOI: https://doi.org/10.1090/S0002-9947-2011-05305-5
Published electronically: July 25, 2011
MathSciNet review: 2833566
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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 $X_p (f,\Lambda )$.


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Additional Information

E. Odell
Affiliation: Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, Texas 78712-0257
Email: odell@math.utexas.edu

B. Sarı
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203-5017
MR Author ID: 741208
Email: bunyamin@unt.edu

Th. Schlumprecht
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
MR Author ID: 260001
Email: schlump@math.tamu.edu

B. Zheng
Affiliation: Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, Texas 78712-0257
Address at time of publication: Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee 38152-3240
Email: btzheng@math.utexas.edu, bzheng@memphis.edu

Received by editor(s): June 10, 2009
Received by editor(s) in revised form: December 25, 2009
Published electronically: July 25, 2011
Additional Notes: The first, third, and fourth authors were partially supported by National Science Foundation Grants DMS-0968813 & DMS-0700126, DMS-0856148 & DMS-0556013, and DMS-1068838, respectively.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.