Gabor frames with atoms in M^q(R) but not in M^p(R) for any 1\leq p < q \leq 2
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| Format: | Preprint |
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2024
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| _version_ | 1866929478617268224 |
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| author | Yu, Pu-Ting |
| author_facet | Yu, Pu-Ting |
| contents | 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 $0<αβ\leq 1$ and $1<q\leq 2$ we explicitly construct Gabor frames $\mathcal{G}(g,α,β)$ 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 $$f = \sum_{k,n\in \mathbb{Z}} \langle f,M_{βn}T_{αk} \mathcal{F}(χ_{[0,α]})\rangle M_{βn}T_{αk} ( \mathcal{F}(χ_{[0,α]}))$$ holds for $f\in M^{p,q}(\mathbb{R})$ with unconditional convergence of the series for any $0<αβ\leq 1$, $1<p<\infty$ 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 $f = \sum_{k,n\in \mathbb{Z}} \langle f, M_{βn}T_{αk} γ\rangle M_{βn}T_{αk} g$ 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 $\{M_{βn}T_{αk}g\}_{n,k\in \mathbb{Z}}$ in $M^{p,q}(\mathbb{R})$ for any $1\leq p,q<\infty$ for which $f = \sum \langle f, γ_{k,n} \rangle M_{βn}T_{αk} g$ with unconditional convergence of the series for all $f$ in $M^{p,q}(\mathbb{R})$ and all alternative duals $\{γ_{k,n}\}_{k,n\in \mathbb{Z}}$ of $\{M_{βn}T_{αk} g\}_{n,k\in \mathbb{Z}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16593 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gabor frames with atoms in M^q(R) but not in M^p(R) for any 1\leq p < q \leq 2 Yu, Pu-Ting Functional Analysis 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 $0<αβ\leq 1$ and $1<q\leq 2$ we explicitly construct Gabor frames $\mathcal{G}(g,α,β)$ 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 $$f = \sum_{k,n\in \mathbb{Z}} \langle f,M_{βn}T_{αk} \mathcal{F}(χ_{[0,α]})\rangle M_{βn}T_{αk} ( \mathcal{F}(χ_{[0,α]}))$$ holds for $f\in M^{p,q}(\mathbb{R})$ with unconditional convergence of the series for any $0<αβ\leq 1$, $1<p<\infty$ 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 $f = \sum_{k,n\in \mathbb{Z}} \langle f, M_{βn}T_{αk} γ\rangle M_{βn}T_{αk} g$ 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 $\{M_{βn}T_{αk}g\}_{n,k\in \mathbb{Z}}$ in $M^{p,q}(\mathbb{R})$ for any $1\leq p,q<\infty$ for which $f = \sum \langle f, γ_{k,n} \rangle M_{βn}T_{αk} g$ with unconditional convergence of the series for all $f$ in $M^{p,q}(\mathbb{R})$ and all alternative duals $\{γ_{k,n}\}_{k,n\in \mathbb{Z}}$ of $\{M_{βn}T_{αk} g\}_{n,k\in \mathbb{Z}}$. |
| title | Gabor frames with atoms in M^q(R) but not in M^p(R) for any 1\leq p < q \leq 2 |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2408.16593 |