A note on the invertibility of the Gabor frame operator on certain modulation spaces

TL;DR

This paper investigates the invertibility of the Gabor frame operator on certain modulation spaces, showing that the dual window remains in the same space if the original window belongs to these spaces.

Contribution

It proves that the Gabor frame operator is bijective on specific modulation spaces, ensuring the dual window shares the same space as the original window.

Findings

Gabor frame operator is bijective on the considered modulation spaces.

The dual window belongs to the same space as the original window.

Results apply to both frames and frame sequences.

Abstract

We consider Gabor frames generated by a general lattice and a window function that belongs to one of the following spaces: the Sobolev space $V_1 = H^1(\mathbb R^d)$, the weighted $L^2$-space $V_2 = L_{1 + |x|}^2(\mathbb R^d)$, and the space $V_3 = \mathbb H^1(\mathbb R^d) = V_1 \cap V_2$ consisting of all functions with finite uncertainty product; all these spaces can be described as modulation spaces with respect to suitable weighted $L^2$ spaces. In all cases, we prove that the space of Bessel vectors in $V_j$ is mapped bijectively onto itself by the Gabor frame operator. As a consequence, if the window function belongs to one of the three spaces, then the canonical dual window also belongs to the same space. In fact, the result not only applies to frames, but also to frame sequences.