Density of Gabor Frames

Abstract A Gabor system is a set of time-frequency shifts S(g, Λ) ={e2 π ibxg(x − a)}(a, b) ∈ Λ of a function g ∈ L2(Rd). We prove that if a finite union of Gabor systems ∪k = 1rS(gk, Λk) forms a frame for L2(Rd) then the lower and upper Beurling densities of Λ = ∪k = 1r Λk satisfy D−(Λ) ≥ 1 and D+ (Λ)

[1]  O. Christensen,et al.  Weyl-Heisenberg frames for subspaces of L^2(R) , 1998, math/9811146.

[2]  Karlheinz Gröchenig,et al.  On Landau's Necessary Density Conditions for Sampling and Interpolation of Band-Limited Functions , 1996 .

[3]  Ole Christensen,et al.  Frames Containing a Riesz Basis and Approximation of the Frame Coefficients Using Finite-Dimensional Methods , 1996 .

[4]  J. Ramanathan,et al.  Incompleteness of Sparse Coherent States , 1995 .

[5]  D. Walnut,et al.  Differentiation and the Balian-Low Theorem , 1994 .

[6]  Ajem Guido Janssen,et al.  Signal Analytic Proofs of Two Basic Results on Lattice Expansions , 1994 .

[7]  H. Landau.  On the density of phase-space expansions , 1993, IEEE Trans. Inf. Theory.

[8]  K. Seip.  Density theorems for sampling and interpolation in the Bargmann-Fock space I. , 1992, math/9204238.

[9]  Ingrid Daubechies,et al.  The wavelet transform, time-frequency localization and signal analysis , 1990, IEEE Trans. Inf. Theory.

[10]  D. Walnut,et al.  Continuous and Discrete Wavelet Transforms , 1989, SIAM Rev..

[11]  M. Rieffel.  Von Neumann algebras associated with pairs of lattices in Lie groups , 1981 .

[12]  L. Hörmander,et al.  An introduction to complex analysis in several variables , 1973 .

[13]  Ronald R. Coifman,et al.  Signal and image representation in combined spaces , 1998 .

[14]  A. Janssen.  A density theorem for time-continuous filter banks , 1998 .

[15]  G. Folland.  Harmonic analysis in phase space , 1989 .

[16]  G. Alexits.  Approximation theory , 1983 .