The existence of Gabor bases and frames

For an arbitrary full rank lattice Λ in R 2d and a function g ∈ L 2 (R d) the Gabor (or Weyl-Heisenberg) system is G(Λ, g): = {e 2πi〈ℓ,x 〉 g(x − κ) ˛ ˛ (κ, ℓ) ∈ Λ}. It is well-known that a necessary condition for G(Λ, g) to be an orthonormal basis for L 2 (R d) is that the density of Λ has D(Λ) = 1. However, except for symplectic lattices it remains an unsolved question whether D(Λ) = 1 is sufficient for the existence of a g ∈ L 2 (R d) such that G(Λ, g) is an orthonormal basis. We investigate this problem and prove that this is true for some of the important cases. In particular we show that this is true for Λ = MZ d where M is either a block triangular matrix or any rational matrix with \ det M \ = 1. Moreover, if M is rational we prove that there exists a compactly supported g such that G(Λ, g) is an orthonormal basis. We also obtain similar results for Gabor frames when D(Λ) ≥ 1. 1.