On the finite linear independence of lattice Gabor systems
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- by Ciprian Demeter and S. Zubin Gautam PDF
- Proc. Amer. Math. Soc. 141 (2013), 1735-1747 Request permission
Abstract:
In the restricted setting of product phase space lattices, we give an alternate proof of P. Linnell’s theorem on the finite linear independence of lattice Gabor systems in $L^2(\mathbb R^d)$. Our proof is based on a simple argument from the spectral theory of random Schrödinger operators; in the one-dimensional setting, we recover the full strength of Linnell’s result for general lattices.References
- Marcin Bownik and Darrin Speegle, Linear independence of Parseval wavelets, Illinois J. Math. 54 (2010), no. 2, 771–785. MR 2846482
- René Carmona and Jean Lacroix, Spectral theory of random Schrödinger operators, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1102675, DOI 10.1007/978-1-4612-4488-2
- Ciprian Demeter, Linear independence of time frequency translates for special configurations, Math. Res. Lett. 17 (2010), no. 4, 761–779. MR 2661178, DOI 10.4310/MRL.2010.v17.n4.a14
- Ciprian Demeter and Alexandru Zaharescu, Proof of the HRT conjecture for $(2,2)$ configurations, J. Math. Anal. Appl. 388 (2012), no. 1, 151–159. MR 2869736, DOI 10.1016/j.jmaa.2011.11.030
- Christopher Heil, Linear independence of finite Gabor systems, Harmonic analysis and applications, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2006, pp. 171–206. MR 2249310, DOI 10.1007/0-8176-4504-7_{9}
- Christopher Heil, Jayakumar Ramanathan, and Pankaj Topiwala, Linear independence of time-frequency translates, Proc. Amer. Math. Soc. 124 (1996), no. 9, 2787–2795. MR 1327018, DOI 10.1090/S0002-9939-96-03346-1
- Svetlana Ya. Jitomirskaya, Metal-insulator transition for the almost Mathieu operator, Ann. of Math. (2) 150 (1999), no. 3, 1159–1175. MR 1740982, DOI 10.2307/121066
- Gitta Kutyniok, Linear independence of time-frequency shifts under a generalized Schrödinger representation, Arch. Math. (Basel) 78 (2002), no. 2, 135–144. MR 1888415, DOI 10.1007/s00013-002-8227-z
- P. A. Linnell, Zero divisors and group von Neumann algebras, Pacific J. Math. 149 (1991), no. 2, 349–363. MR 1105703, DOI 10.2140/pjm.1991.149.349
- Peter A. Linnell, von Neumann algebras and linear independence of translates, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3269–3277. MR 1637388, DOI 10.1090/S0002-9939-99-05102-3
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Christoph Thiele, 2007, Personal communication.
Additional Information
- Ciprian Demeter
- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- MR Author ID: 734783
- Email: demeterc@indiana.edu
- S. Zubin Gautam
- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- Address at time of publication: School of Law, University of California, Berkeley, California 94720
- Email: sgautam@indiana.edu, sgautam@berkeley.edu
- Received by editor(s): December 26, 2010
- Received by editor(s) in revised form: September 13, 2011
- Published electronically: November 29, 2012
- Additional Notes: The first author is supported by a Sloan Research Fellowship and by NSF Grant DMS-0901208.
- Communicated by: Michael T. Lacey
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 1735-1747
- MSC (2010): Primary 42C40, 42B99, 26B99; Secondary 46B15
- DOI: https://doi.org/10.1090/S0002-9939-2012-11452-2
- MathSciNet review: 3020859