Invertibility of the Gabor frame operator on the Wiener amalgam space

TL;DR

This paper proves that Gabor frame operators with generators in a specific Wiener amalgam space are invertible on that space, ensuring the dual generator also belongs to the same space, using a generalized Wiener's theorem.

Contribution

It extends Wiener's $1/f$ theorem to show invertibility of Gabor frame operators on Wiener amalgam spaces, establishing dual generators within the same function space.

Findings

Gabor frame operator is invertible on Wiener amalgam space for generators in that space.

The canonical dual generator also belongs to the same Wiener amalgam space.

The proof uses a generalized version of Wiener's $1/f$ theorem.

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 $W(L^{\infty}, \ell^{1}_{\nu})(\mathbb{R}^{d})$, 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^{1}_{\nu})(\mathbb{R}^{d}) $