Duality of Gabor frames and Heisenberg modules
Duality of Gabor frames and Heisenberg modules
复制标题
Gabor框架和海森堡模块的对偶性
DOI：
--
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
F. Luef
中科院分区：
文献类型：
--
作者：
Mads S. Jakobsen;F. Luef
Given a locally compact abelian group $G$ and a closed subgroup $Lambda$ in $G imeswidehat{G}$, Rieffel associated to $Lambda$ a Hilbert $C^*$-module $mathcal{E}$, known as a Heisenberg module. He proved that $mathcal{E}$ is an equivalence bimodule between the twisted group $C^*$-algebra $C^*(Lambda, extsf{c})$ and $C^*(Lambda^circ,ar{ extsf{c}})$, where $Lambda^{circ}$ denotes the adjoint subgroup of $Lambda$. Our main goal is to study Heisenberg modules using tools from time-frequency analysis and pointing out that Heisenberg modules provide the natural setting of the duality theory of Gabor systems. More concretely, we show that the Feichtinger algebra ${ extbf{S}}_{0}(G)$ is an equivalence bimodule between the Banach subalgebras ${ extbf{S}}_{0}(Lambda, extsf{c})$ and ${ extbf{S}}_{0}(Lambda^{circ},ar{ extsf{c}})$ of $C^*(Lambda, extsf{c})$ and $C^*(Lambda^circ,ar{ extsf{c}})$, respectively. Further, we prove that ${ extbf{S}}_{0}(G)$ is finitely generated and projective exactly for co-compact closed subgroups $Lambda$. In this case the generators $g_1,ldots,g_n$ of the left ${ extbf{S}}_{0}(Lambda)$-module ${ extbf{S}}_{0}(G)$ are the Gabor atoms of a multi-window Gabor frame for $L^2(G)$. We prove that this is equivalent to $g_1,ldots,g_n$ being a Gabor super frame for the closed subspace generated by the Gabor system for $Lambda^{circ}$. This duality principle is of independent interest and is also studied for infinitely many Gabor atoms. We also show that for any non-rational lattice $Lambda$ in $mathbb{R}^{2m}$ with volume ${s}(Lambda)<1$ there exists a Gabor frame generated by a single atom in ${ extbf{S}}_{0}(mathbb{R}^m)$.
Given a locally compact abelian group $G$ and a closed subgroup $Lambda$ in $G imeswidehat{G}$, Rieffel associated to $Lambda$ a Hilbert $C^*$-module $mathcal{E}$, known as a Heisenberg module. He proved that $mathcal{E}$ is an equivalence bimodule between the twisted group $C^*$-algebra $C^*(Lambda, extsf{c})$ and $C^*(Lambda^circ,ar{ extsf{c}})$, where $Lambda^{circ}$ denotes the adjoint subgroup of $Lambda$. Our main goal is to study Heisenberg modules using tools from time-frequency analysis and pointing out that Heisenberg modules provide the natural setting of the duality theory of Gabor systems. More concretely, we show that the Feichtinger algebra ${ extbf{S}}_{0}(G)$ is an equivalence bimodule between the Banach subalgebras ${ extbf{S}}_{0}(Lambda, extsf{c})$ and ${ extbf{S}}_{0}(Lambda^{circ},ar{ extsf{c}})$ of $C^*(Lambda, extsf{c})$ and $C^*(Lambda^circ,ar{ extsf{c}})$, respectively. Further, we prove that ${ extbf{S}}_{0}(G)$ is finitely generated and projective exactly for co-compact closed subgroups $Lambda$. In this case the generators $g_1,ldots,g_n$ of the left ${ extbf{S}}_{0}(Lambda)$-module ${ extbf{S}}_{0}(G)$ are the Gabor atoms of a multi-window Gabor frame for $L^2(G)$. We prove that this is equivalent to $g_1,ldots,g_n$ being a Gabor super frame for the closed subspace generated by the Gabor system for $Lambda^{circ}$. This duality principle is of independent interest and is also studied for infinitely many Gabor atoms. We also show that for any non-rational lattice $Lambda$ in $mathbb{R}^{2m}$ with volume ${s}(Lambda)<1$ there exists a Gabor frame generated by a single atom in ${ extbf{S}}_{0}(mathbb{R}^m)$.
影响因子：
1.2
作者：
Balan, R.;Dutkay, D.;Han, D.;Larson, D.;Luef, F.
通讯作者：
Luef, F.