Gabor Frames without Inequalities

We prove fourteen equivalent conditions for a set of time-frequency shifts on a lattice Λ, {e2g(t − λ1) : (λ1, λ2) ∈ Λ} ⊆ L (R), to be a frame for L(R). Remarkably, several of these conditions can be formulated without an inequality. In particular, instead of checking the invertibility of the frame operator on L(R), it suffices to verify that it is one-to-one on a certain subspace of tempered distributions.