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 $0<\alpha \beta \leq 1$ 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 $p\leq q\leq p'$ 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 $\{M_{\beta n}T_{\alpha k} g\}_{n,k\in \mathbb{Z}}$.