For a second countable locally compact abelian (LCA) group $G$, we study some necessary and sufficient conditions to generate continuous Gabor frames for $L^{2}(G)$. To this end, we reformulate the generalized Zak transform proposed by Grochenig in the case of integer-oversampled lattices, however our formulation rely on the assumption that both translation and modulation groups are only closed subgroups. Moreover, we discuss the possibility of such generalization and apply several examples to demonestrate the necessity of standing conditions in the results. Finally, by using the generalized Zak transform and fiberization technique, we obtain some characterization of continuous Gabor frames for $L^{2}(G)$ in term of a family of frames in $l^{2}(\widehat{H^{\perp}})$ for a closed co-compact subgroup $H$ of $G$.
We generalize Feichtinger and Kaiblinger's theorem on linear deformations of uniform Gabor frames to the setting of a locally compact abelian group $G$. More precisely, we show that Gabor frames over lattices in the time-frequency plane of $G$ with windows in the Feichtinger algebra are stable under small deformations of the lattice by an automorphism of $G \times \widehat{G}$. The topology we use on the automorphisms is the Braconnier topology. We characterize the groups in which the Balian--Low theorem for the Feichtinger algebra holds as exactly the groups with noncompact identity component…
G\v avruta studied atomic systems in terms of frames for range of operators (that is, for subspaces), namely $K$-frames, where the lower frame condition is controlled by the Hilbert-adjoint of a bounded linear operator $K$. For a locally compact abelian group G and a positive integer $n$, we study frames of matrix-valued Gabor systems in the matrix-valued Lebesgue space $L^2(G, \mathbb{C}^{n\times n})$ , where a bounded linear operator $Θ$ on $L^2(G, \mathbb{C}^{n\times n})$ controls not only lower but also the upper frame condition. We term such frames matrix-valued $(Θ, Θ^*)$-Gabor frames. F…
We introduce new quasi-Banach modulation spaces on locally compact abelian (LCA) groups which coincide with the classical ones in the Banach setting and prove their main properties. Then we study Gabor frames on quasi-lattices, significantly extending the original theory introduced by Gröchenig and Strohmer. These issues are the key tools in showing boundedness results for Kohn-Nirenberg and localization operators on modulation spaces and studying their eigenfunctions' properties. In particular, the results in the Euclidean space are recaptured.
Given a locally compact abelian group $G$ and a closed subgroup $Λ$ in $G\times\widehat{G}$, Rieffel associated to $Λ$ 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 $C^*(Λ,\textsf{c})$ and $C^*(Λ^\circ,\bar{\textsf{c}})$, where $Λ^{\circ}$ denotes the adjoint subgroup of $Λ$. 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 concr…
Let $Λ$ be a lattice in a second countable, locally compact abelian group $G$ with annihilator $Λ^{\perp} \subseteq \widehat{G}$. We investigate the validity of the following statement: For every $η$ in the Feichtinger algebra $S_0(G)$, the Gabor system $\{ M_τ T_λ η\}_{λ\in Λ, τ\in Λ^{\perp}}$ is not a frame for $L^2(G)$. When $G = \mathbb{R}$ and $Λ= α\mathbb{Z}$, this statement is a variant of the Balian-Low theorem. Extending a result of R. Balan, we show that whether the statement generalizes to $(G,Λ)$ is equivalent to the nontriviality of a certain vector bundle over the compact space $…
By using a coset of closed subgroup, we define a Fourier like transform for locally compact abelian (LCA) topological groups. We studied two wavelet multipliers and Landau-Pollak-Slepian operators on locally compact abelian topological groups associated to the transform and show that the transforms are $L^p$bounded linear operators, and are in Schatten p-class for $1\leq p\leq \infty$. Finally, we determine their trace class and also obtain a connection with the generalized Landau-Pollak-Slepian operators.
Functional Models of Abelian Locally von Neumann Algebras and Direct Integrals of Locally Hilbert Spaces
Institute of Mathematics of the Romanian Academy（罗马尼亚科学院数学研究所）；Bilkent University（比尔肯特大学）；Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Delhi Centre（印度统计学院德里中心理论与统计数学单元）；Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali（印度科学教育与研究学院（IISER）莫哈利数学科学系）
本文研究了可积局部 von Neumann 代数的功能模型，通过局部本质有界函数构建，探讨了局部 Hilbert 空间直接积分及局部可分解与可对角化的算子，证明其构成互为对易的局部 von Neumann 代数。
Akbarov's theory of holomorphic reflexivity for topological Hopf algebras has been developed in two directions, namely, by the complication of definitions when expanding the scope and by their simplification when restricting. In the framework of the latter approach, we establish the holomorphic reflexivity for topological Hopf algebras associated with locally finite countable groups and second-countable profinite groups. In the Abelian case, the reflexivity is described in terms close to the classical ones.
