Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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)$.


References [Enhancements On Off] (What's this?)

  • [AOd] D. Alspach and E. Odell, ``$ L_p$ spaces'', in Handbook of Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds., Elsevier, Amsterdam (2001), 123-159. MR 1863691 (2003b:46013)
  • [AO] A. Atzmon and A. Olevskii, Completeness of integer translates in function spaces on $ \mathbb{R}$, J. of Approx. Theory 87 (1996), 291-327. MR 1420335 (98a:46035)
  • [BOU] J. Bruna, A. Olevskii and A. Ulanovskii, Completeness in $ L_1(\mathbb{R})$ of discrete translates, Rev. Mat. Iberoamericana 22 no. 1, (2006), 1-16. MR 2267311 (2007k:42015)
  • [CDOSZ] P. G. Casazza, S. J. Dilworth, E. Odell, Th. Schlumprecht and A. Zsák, Coefficient quantization for frames in Banach spaces, J. Math. Anal. Appl. 348 (2008), 66-86. MR 2449328 (2009g:46018)
  • [CHL] P. G. Casazza, D. Han and D. R. Larson, Frames for Banach spaces, the functional and harmonic analysis of wavelets and frames, (San Antonio, TX, 1999), Contemp. Math. 247 (1999), 149-182. MR 1738089 (2000m:46015)
  • [DH] B. Deng and C. Heil, ``Density of Gabor Schauder bases'', in Wavelet Applications in Signal and Image Processing VIII (San Diego, CA, 2000), A. Aldroubi, A. Lane, and M. Unser, eds., Proc. SPIE 4119, SPIE, Bellingham, WA, (2000), 153-164.
  • [ER] G. Edgar and J. Rosenblatt, Difference equations over locally compact abelian groups, Trans. Amer. Math. Soc. 253 (1979), 273-289. MR 536947 (80i:39001)
  • [H] C. Heil, ``Linear independence of finite Gabor systems'' in Harmonic Analysis and Applications, A volume in honor of John J. Benedetto, Birkhauser, Boston, (2006), 171-206. MR 2249310 (2007d:42057)
  • [J] W. B. Johnson, On quotients of $ L_p$ which are quotients of $ \ell_p$, Compositio Math. 34 (1) (1977), 69-89. MR 0454595 (56:12844)
  • [JL] W. B. Johnson and J. Lindenstrauss, ``Basic concepts in the geometry of Banach spaces'', in Handbook of Geometry of Banach Spaces, Vol. 1, W. B. Johnson and J. Lindenstrauss, eds., Elsevier, Amsterdam (2001), 1-84. MR 1863689 (2003f:46013)
  • [JO] W. B. Johnson and E. Odell, Subspaces of $ L_p$ which embed into $ \ell_p$, Compos. Math. 28 (1974), 37-49. MR 0352938 (50:5424)
  • [JMST] W. B. Johnson, B. Maurey, G. Schechtman and L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 19 (217) (1979) v+298. MR 527010 (82j:46025)
  • [KP] M. I. Kadets and A. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces $ L_p$, Studia Math. 21 (1961/62), 161-176. MR 0152879 (27:2851)
  • [KW] N. J. Kalton and D. Werner, Property $ (M)$, $ M$-ideals, and almost isometric structure of Banach spaces, J. Reine Angew. Math. 461 (1995), 137-178. MR 1324212 (96m:46022)
  • [Ko] P. Koosis, The logarithmic integral II, Cambridge Studies in Advanced Mathematics, 21. Cambridge University Press, Cambridge, 1992. MR 1195788 (94i:30027)
  • [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 92, Springer-Verlag, Berlin (1977). MR 0500056 (58:17766)
  • [Ol] A. Olevskii, Completeness in $ L_2(\mathbb{R})$ of almost integer translates, C. R. Acad. Sci. Paris 324 (1997), 987-991. MR 1451238 (98a:42002)
  • [OZ] T. E. Olson and R. A. Zalik, ``Nonexistence of a Riesz basis of translates'', Approximation Theory, Lecture Notes in Pure and Applied Math., 138, Dekker, New York (1992) 401-408. MR 1174120
  • [R] J. Rosenblatt, private communication.
  • [Ro] J. Rosenblatt, Linear independence of translations, J. Austral. Math. Soc. (Series A) 59 (1995), 131-133. MR 1336456 (96f:42009)
  • [Ru] W. Rudin, Fourier Analysis on Groups, Interscience Publisher, John Wiley & Sons (1967). MR 0152834 (27:2808)
  • [S] G. Schechtman, A remark on unconditional basic sequences in $ L_p$ $ (1<p<\infty)$, Israel J. Math. 19 (1974), 220-224. MR 0511797 (58:23512)
  • [Wi] N. Wiener, The Fourier integral and certain of its applications, Cambridge University Press, Cambridge, 1933 reprint: Dover, New York, 1958. MR 0100201 (20:6634)
  • [Yo] R. M. Young, An introduction to nonharmonic Fourier series, Pure and Applied Mathematics, 93. Academic Press, Inc., New York-London, 1980. MR 591684 (81m:42027)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 42C30, 46E30, 46B15

Retrieve articles in all journals with MSC (2010): 42C30, 46E30, 46B15


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: https://doi.org/10.1090/S0002-9947-2011-05305-5
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.

American Mathematical Society