This paper consists of two parts. In the first half, we solve the question raised by Heil as to whether the atom of a Gabor frame must be in \(M^p(\mathbb{R})\) for some \(1 < p < 2\) . Specifically, for each \(0 < \alpha \beta \leq 1\) and \(1 < q\leq 2\) we explicitly construct Gabor frames \(\mathcal{G}(g,\alpha,\beta)\) with atoms in \(M^q(\mathbb{R})\) but not in \(M^{p}(\mathbb{R})\) for any \(1\leq p < q\) . To construct such Gabor frames, we use box functions as the window functions and show that \(f = \sum_{k,n\in \mathbb{Z}}\langle\,f,M_{\beta n}T_{\alpha k} {\mathcal{F}}(\chi_{[0,\alpha]})\,\rangle M_{\beta n}T_{\alpha k} ( {\mathcal{F}}(\chi_{[0,\alpha]}))\) holds for \(f\in M^{p,q}(\mathbb{R})\) with unconditional convergence of the series for any \(0 < \alpha\beta \leq 1\) , \(1 < p < \infty\) and \(1\leq q < \infty\) .
In the second half of this paper, we study two questions related to unconditional convergence of Gabor expansions in modulation spaces. Under the assumption that the window functions are chosen from \(M^p(\mathbb{R})\) for some \(1\leq p\leq 2,\) we will prove several equivalent statements that the equation \(f = \sum_{k,n\in \mathbb{Z}} \langle\,f,M_{\beta n}T_{\alpha k} \gamma\,\rangle M_{\beta n}T_{\alpha k} g\) can be extended from \(L^2(\mathbb{R})\) to \(M^q(\mathbb{R})\) for all \(f\in M^q(\mathbb{R})\) and all \(p\leq q\leq p^{\prime}\) with unconditional convergence of the series. Finally, we characterize all Gabor systems \(\{M_{\beta n}T_{\alpha k} g\}_{n,k\in \mathbb{Z}}\) in \(M^{p,q}(\mathbb{R})\) for any \(1\leq p,q < \infty\) for which \(f = \sum\langle\,f,\gamma_{k,n}\,\rangle M_{\beta n}T_{\alpha k} g\) with unconditional convergence of the series for all f in \(M^{p,q}(\mathbb{R})\) and all alternative duals \(\{\gamma_{k,n}\}_{k,n\in \mathbb{Z}}\) of \(\{M_{\beta n}T_{\alpha k} g\}_{n,k\in \mathbb{Z}}\) .
