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Transactions of the American Mathematical Society
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
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
Email: bunyamin@unt.edu

Th. Schlumprecht
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
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

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05305-5
PII: S 0002-9947(2011)05305-5
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.