Multi-window Gabor frames in amalgam spaces

TL;DR

This paper demonstrates that multi-window Gabor frames with windows in the Wiener algebra serve as Banach frames for Wiener amalgam spaces, resolving an open question about the continuity of their duals using Wiener's lemma.

Contribution

It establishes the Banach frame property for multi-window Gabor frames in Wiener amalgam spaces and confirms the continuity of their duals, answering an open problem.

Findings

Multi-window Gabor frames are Banach frames for Wiener amalgam spaces.

The canonical dual of a Gabor frame with a continuous generator is continuous.

The results are proved using Wiener's $1/f$ lemma.

Abstract

We show that multi-window Gabor frames with windows in the Wiener algebra $W(L^{\infty}, \ell^{1})$ are Banach frames for all Wiener amalgam spaces. As a byproduct of our results we positively answer an open question that was posed by [Krishtal and Okoudjou, Invertibility of the Gabor frame operator on the Wiener amalgam space, J. Approx. Theory, 153(2), 2008] and concerns the continuity of the canonical dual of a Gabor frame with a continuous generator in the Wiener algebra. The proofs are based on a recent version of Wiener's $1/f$ lemma.