Invertibility of the Gabor frame operator on the Wiener amalgam space

We use a generalization of Wiener's 1/f theorem to prove that for a Gabor frame with the generator in the Wiener amalgam space W(L∞,ℓ1)(Rd), the corresponding frame operator is invertible on this space. Therefore, for such a Gabor frame, the canonical dual belongs also to W(L∞,ℓ1)(Rd).