Invertibility of the Gabor frame operator on the Wiener amalgam space

Abstract

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 $W(L^{\infty },{\ell }_{\nu }^1)({\mathbb{R}}^d)$, the corresponding frame operator is invertible on this space. Therefore, for such a Gabor frame, the generator of the canonical dual belongs also to $W(L^{\infty },{\ell }_{\nu }^1)({\mathbb{R}}^d)$