Gabor frames with atoms in M^q(R) but not in M^p(R) for any 1\leq p < q \leq 2

Abstract

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 $0<\alpha \beta \le 1$ and $1<q\le 2$ 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 $1\le p<q$. To construct such Gabor frames, we use box functions as the window functions and show that $$f=\sum _{k,n\in \mathbb{Z}}⟨f,M_{\beta n}{T}_{\alpha k}\mathcal{F}({\chi }_{[0,\alpha ]})⟩M_{\beta n}{T}_{\alpha k}(\mathcal{F}({\chi }_{[0,\alpha ]}))$$ holds for $f\in M^{p,q}(\mathbb{R})$ with unconditional convergence of the series for any $0<\alpha \beta \le 1$, $1<p<\infty$ and $1\le q<\infty$. 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 $1\le p\le 2,$ we will prove several equivalent statements that the equation $f={\sum }_{k,n\in \mathbb{Z}}⟨f,M_{\beta n}{T}_{\alpha k}\gamma ⟩M_{\beta n}{T}_{\alpha k}g$ can be extended from $L^2(\mathbb{R})$ to $M^q(\mathbb{R})$ for all $f\in M^q(\mathbb{R})$ and all $p\le q\le p^{\prime }$ 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 $1\le p,q<\infty$ for which $f=\sum ⟨f,{\gamma }_{k,n}⟩M_{\beta n}{T}_{\alpha k}g$ 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 $\{M_{\beta n}{T}_{\alpha k}g{\}}_{n,k\in \mathbb{Z}}$.

Any comment on this topic or the manuscript would be greatly appreciated