A note on ℓ2-linear independence of Gabor systems.

It is known that the system of integer translates of ψ ∈ L 2 (R) \ { 0 } is ℓ 2 (Z) -linearly independent precisely when the corresponding periodization function is strictly positive almost everywhere. The only known proof of this result is based on certain properties of the space of functions with uniformly bounded Fourier partial sums. We show that one of these properties does not hold for the analogue of this space in higher dimensions and square summation and, therefore, this method cannot be applied to solve the problem for integer translates of function ψ ∈ L 2 (R d) for d ⩾ 2 or ℓ 2 (Z 2 d) -linear independence of Gabor systems { T k M ℓ ψ : k , ℓ ∈ Z 2 d } for ψ ∈ L 2 (R d) , with d ⩾ 1 . [ABSTRACT FROM AUTHOR]