{"title":"对于任意 1\\leq p < q \\leq 2，原子在 M^q(R)中而不在 M^p(R)中的 Gabor 框架","authors":"Pu-Ting Yu","doi":"arxiv-2408.16593","DOIUrl":null,"url":null,"abstract":"This paper consists of two parts. In the first half, we solve the question\nraised by Heil as to whether the atom of a Gabor frame must be in\n$M^p(\\mathbb{R})$ for some $1<p<2$. Specifically, for each $0<\\alpha \\beta \\leq 1$ and $1<q\\leq 2$ we explicitly construct\nGabor frames $\\mathcal{G}(g,\\alpha,\\beta)$ with atoms in $M^q(\\mathbb{R})$ but\nnot in $M^{p}(\\mathbb{R})$ for any $1\\leq p<q$. To construct such Gabor frames,\nwe use box functions as the window functions and show that $$f = \\sum_{k,n\\in\n\\mathbb{Z}} \\langle f,M_{\\beta n}T_{\\alpha k}\n\\mathcal{F}(\\chi_{[0,\\alpha]})\\rangle M_{\\beta n}T_{\\alpha k} (\n\\mathcal{F}(\\chi_{[0,\\alpha]}))$$ holds for $f\\in M^{p,q}(\\mathbb{R})$ with\nunconditional convergence of the series for any $0<\\alpha\\beta \\leq 1$,\n$1<p<\\infty$ and $1\\leq q<\\infty$. In the second half of this paper, we study two questions related to\nunconditional convergence of Gabor expansions in modulation spaces. Under the\nassumption that the window functions are chosen from $M^p(\\mathbb{R})$ for some\n$1\\leq p\\leq 2,$ we will prove several equivalent statements that the equation\n$f = \\sum_{k,n\\in \\mathbb{Z}} \\langle f, M_{\\beta n}T_{\\alpha k} \\gamma \\rangle\nM_{\\beta n}T_{\\alpha k} g$ can be extended from $L^2(\\mathbb{R})$ to\n$M^q(\\mathbb{R})$ for all $f\\in M^q(\\mathbb{R})$ and all $p\\leq q\\leq p'$ with\nunconditional convergence of the series. Finally, we characterize all Gabor\nsystems $\\{M_{\\beta n}T_{\\alpha k}g\\}_{n,k\\in \\mathbb{Z}}$ in\n$M^{p,q}(\\mathbb{R})$ for any $1\\leq p,q<\\infty$ for which $f = \\sum \\langle f,\n\\gamma_{k,n} \\rangle M_{\\beta n}T_{\\alpha k} g$ with unconditional convergence\nof the series for all $f$ in $M^{p,q}(\\mathbb{R})$ and all alternative duals\n$\\{\\gamma_{k,n}\\}_{k,n\\in \\mathbb{Z}}$ of $\\{M_{\\beta n}T_{\\alpha k}\ng\\}_{n,k\\in \\mathbb{Z}}$.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Gabor frames with atoms in M^q(R) but not in M^p(R) for any 1\\\\leq p < q \\\\leq 2\",\"authors\":\"Pu-Ting Yu\",\"doi\":\"arxiv-2408.16593\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper consists of two parts. In the first half, we solve the question\\nraised by Heil as to whether the atom of a Gabor frame must be in\\n$M^p(\\\\mathbb{R})$ for some $1<p<2$. Specifically, for each $0<\\\\alpha \\\\beta \\\\leq 1$ and $1<q\\\\leq 2$ we explicitly construct\\nGabor frames $\\\\mathcal{G}(g,\\\\alpha,\\\\beta)$ with atoms in $M^q(\\\\mathbb{R})$ but\\nnot in $M^{p}(\\\\mathbb{R})$ for any $1\\\\leq p<q$. To construct such Gabor frames,\\nwe use box functions as the window functions and show that $$f = \\\\sum_{k,n\\\\in\\n\\\\mathbb{Z}} \\\\langle f,M_{\\\\beta n}T_{\\\\alpha k}\\n\\\\mathcal{F}(\\\\chi_{[0,\\\\alpha]})\\\\rangle M_{\\\\beta n}T_{\\\\alpha k} (\\n\\\\mathcal{F}(\\\\chi_{[0,\\\\alpha]}))$$ holds for $f\\\\in M^{p,q}(\\\\mathbb{R})$ with\\nunconditional convergence of the series for any $0<\\\\alpha\\\\beta \\\\leq 1$,\\n$1<p<\\\\infty$ and $1\\\\leq q<\\\\infty$. In the second half of this paper, we study two questions related to\\nunconditional convergence of Gabor expansions in modulation spaces. Under the\\nassumption that the window functions are chosen from $M^p(\\\\mathbb{R})$ for some\\n$1\\\\leq p\\\\leq 2,$ we will prove several equivalent statements that the equation\\n$f = \\\\sum_{k,n\\\\in \\\\mathbb{Z}} \\\\langle f, M_{\\\\beta n}T_{\\\\alpha k} \\\\gamma \\\\rangle\\nM_{\\\\beta n}T_{\\\\alpha k} g$ can be extended from $L^2(\\\\mathbb{R})$ to\\n$M^q(\\\\mathbb{R})$ for all $f\\\\in M^q(\\\\mathbb{R})$ and all $p\\\\leq q\\\\leq p'$ with\\nunconditional convergence of the series. Finally, we characterize all Gabor\\nsystems $\\\\{M_{\\\\beta n}T_{\\\\alpha k}g\\\\}_{n,k\\\\in \\\\mathbb{Z}}$ in\\n$M^{p,q}(\\\\mathbb{R})$ for any $1\\\\leq p,q<\\\\infty$ for which $f = \\\\sum \\\\langle f,\\n\\\\gamma_{k,n} \\\\rangle M_{\\\\beta n}T_{\\\\alpha k} g$ with unconditional convergence\\nof the series for all $f$ in $M^{p,q}(\\\\mathbb{R})$ and all alternative duals\\n$\\\\{\\\\gamma_{k,n}\\\\}_{k,n\\\\in \\\\mathbb{Z}}$ of $\\\\{M_{\\\\beta n}T_{\\\\alpha k}\\ng\\\\}_{n,k\\\\in \\\\mathbb{Z}}$.\",\"PeriodicalId\":501036,\"journal\":{\"name\":\"arXiv - MATH - Functional Analysis\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Functional Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.16593\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Functional Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.16593","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Gabor frames with atoms in M^q(R) but not in M^p(R) for any 1\leq p < q \leq 2
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
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