On the finite linear independence of lattice Gabor systems

Demeter, Ciprian; Gautam, S.

doi:10.1090/S0002-9939-2012-11452-2

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.