Figure 1

The Gabor frame set. The figure shows the following regions: B as in Proposition 1.1(iii), and D as in Theorem 1.2. The region A corresponds to the case where the frame operator is a multiplication operator, and \(\{E_{mb}T_{na}g \}_{m,n \in \mathbb {Z}}\) is a frame if \(\inf_{x\in [0,a]} \sum_{n\in \mathbb {Z}} |g(x-na)|^{2} >0\). The region C is a part of the region determined by Proposition 1.1(ii), corresponding to the new findings in the paper [21].