A Note on the Invertibility of the Gabor Frame Operator on Certain Modulation Spaces

Abstract We consider Gabor frames generated by a general lattice and window function that belongs to one of the following spaces: Sobolev space $$V_1 = H^1(\mathbb {R}^d)$$ V 1 = H ( R d ) , weighted $$L^2$$ L 2 -space $$V_2 L_{1 + |x|}^2(\mathbb + | x $$V_3 \mathbb {H}^1(\mathbb {R}^d) V_1 \cap V_2$$ 3 ∩ consisting all functions with finite uncertainty product; these spaces can be described as modulation respect suitable spaces. In cases, we prove Bessel vectors in $$V_j$$ j is mapped bijectively onto itself frame operator. As consequence, if three spaces, then canonical dual also same space. fact, result not only applies frames, but sequences.