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Register a new userOrbits of bounded bijective operators and Gabor frames
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Rosario Corso
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This paper is a contribution to frame theory. Frames in a Hilbert space are generalizations of orthonormal bases. In particular, Gabor frames of $L^2(mathbb{R})$, which are made of translations and modulations of one or more windows, are often used in applications. More precisely, the paper deals with a question posed in the last years by Christensen and Hasannasab about the existence of overcomplete Gabor frames, with some ordering over $mathbb{Z}$, which are orbits of bounded operators on $L^2(mathbb{R})$. Two classes of overcomplete Gabor frames which cannot be ordered over $mathbb{Z}$ and represented by orbits of operators in $GL(L^2(mathbb{R}))$ are given. Some results about operator representation are stated in a general context for arbitrary frames, covering also certain wavelet frames.
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Given a locally compact abelian group $G$ and a closed subgroup $Lambda$ in $Gtimeswidehat{G}$, 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 $C^*(Lambda,textsf{c})$ and $C^*(Lambda^circ,bar{textsf{c}})$, 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 ${textbf{S}}_{0}(G)$ is an equivalence bimodule between the Banach subalgebras ${textbf{S}}_{0}(Lambda,textsf{c})$ and ${textbf{S}}_{0}(Lambda^{circ},bar{textsf{c}})$ of $C^*(Lambda,textsf{c})$ and $C^*(Lambda^circ,bar{textsf{c}})$, respectively. Further, we prove that ${textbf{S}}_{0}(G)$ 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 ${textbf{S}}_{0}(Lambda)$-module ${textbf{S}}_{0}(G)$ 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 ${textbf{S}}_{0}(mathbb{R}^m)$.
The main purpose of the paper is to give a characterization of all compactly supported dual windows of a Gabor frame. As an application, we consider an iterative procedure for approximation of the canonical dual window via compactly supported dual windows on every step. In particular, the procedure allows to have approximation of the canonical dual window via dual windows from certain modulation spaces or from the Schwartz space.
In 2012 Gu{a}vruc{t}a introduced the notions of $K$-frame and of atomic system for a linear bounded operator $K$ in a Hilbert space $mathcal{H}$, in order to decompose its range $mathcal{R}(K)$ with a frame-like expansion. In this article we revisit these concepts for an unbounded and densely defined operator $A:mathcal{D}(A)tomathcal{H}$ in two different ways. In one case we consider a non-Bessel sequence where the coefficient sequence depends continuously on $finmathcal{D}(A)$ with respect to the norm of $mathcal{H}$. In the other case we consider a Bessel sequence and the coefficient sequence depends continuously on $finmathcal{D}(A)$ with respect to the graph norm of $A$.
In the past decade, significant progress has been made to generalize classical tools from Fourier analysis to analyze and process signals defined on networks. In this paper, we propose a new framework for constructing Gabor-type frames for signals on graphs. Our approach uses general and flexible families of linear operators acting as translations. Compared to previous work in the literature, our methods yield the sharp bounds for the associated frames, in a broad setting that generalizes several existing constructions. We also examine how Gabor-type frames behave for signals defined on Cayley graphs by exploiting the representation theory of the underlying group. We explore how natural classes of translations can be constructed for Cayley graphs, and how the choice of an eigenbasis can significantly impact the properties of the resulting translation operators and frames on the graph.
We generalize Feichtinger and Kaiblingers 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. This generalizes a theorem of Kaniuth and Kutyniok on the zeros of the Zak transform on locally compact abelian groups. We apply our results to a class of number-theoretic groups, including the adele group associated to a global field.
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