We investigate vector-valued Gabor frames (sometimes called Gabor superframes) based on Hermite functions $H_n$. Let $h= (H_0, H_1, ..., H_n)$ be the vector of the first $n+1$ Hermite functions. We give a complete characterization of all lattices $Λ\subseteq \bR ^2$ such that the Gabor system
$\{e^{2πi λ_2 t} \boh (t-λ_1): λ= (λ_1, λ_2) \in Λ\}$ is a frame for $L^2 (\bR, \bC ^{n+1})$. As a corollary we obtain sufficient conditions for a single Hermite function to generate a Gabor frame and a new estimate for the lower frame bound.
The main tools are growth estimates for the Weierstrass $σ$-function, a new type of interpolation problem for entire functions on the Bargmann-Fock space, and structural results about vector-valued Gabor frames.
Gabor frames with Hermite functions are equivalent to sampling sequences in true Fock spaces of polyanalytic functions. In the L^2-case, such an equivalence follows from the unitarity of the polyanalytic Bargmann transform. We will introduce Banach spaces of polyanalytic functions and investigate the mapping properties of the polyanalytic Bargmann transform on modulation spaces. By applying the theory of coorbit spaces and localized frames to the Fock representation of the Heisenberg group, we derive explicit polyanalytic sampling theorems which can be seen as a polyanalytic version of the lat…
We prove some infinite series identities for the Hermite functions. From these identities we disprove the Gabor frame set conjecture for Hermite functions of order $4m+2$ and $4m+3$ for $m \in \{0\} \cup \mathbb{N}$. The results hold not only for Hermite functions, but for two large classes of eigenfunctions of the Fourier transform associated with the eigenvalues $-1$ and $i$, and the results indicate that the Gabor frame set of all such functions must have a rather complicated structure.
We derive frame estimates for vector-valued Gabor systems with window functions belonging to Schwartz space. The main result provides frame bound estimates for windows composed of Hermite functions. The proof is based on a recently established sampling theorem for the simply connected Heisenberg group, which is translated to a family of frame estimates via a direct integral decomposition.
Asymptotic safety regions for Gabor frames generated by Hermite functions
University of Vienna（维也纳大学）；Norwegian University of Science and Technology (NTNU)（挪威科技大学）；Erwin Schrödinger International Institute for Mathematics and Physics (ESI), University of Vienna（维也纳大学埃尔温·薛定谔数学与物理国际研究所）；King’s College London（伦敦国王学院）
Using Gabor analysis, we give a complete characterization of all lattice sampling and interpolating sequences in the Fock space of polyanalytic functions, displaying a "Nyquist rate" which increases with $n$, the degree of polyanaliticity of the space. Such conditions are equivalent to sharp lattice density conditions for certain vector-valued Gabor systems, namely superframes and Gabor super-Riesz sequences with Hermite windows, and in the case of superframes they were studied recently by Gröchenig and Lyubarskii. The proofs of our main results use variations of the Janssen-Ron-Shen duality p…
In this work we deal with the recently introduced concept of weaving frames. We extend the concept to include multi-window frames and present the first sufficient criteria for a family of multi-window Gabor frames to be woven. We give a Hilbert space norm criterion and a pointwise criterion in phase space. The key ingredient are localization operators in phase space and we give examples of woven multi-window Gabor frames consisting of Hermite functions.
The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalent $C^*$-algebras where the equivalence bimodule is a finitely generated projective Hilbert $C^*$-module. These Hilbert $C^*$-modules are equipped with some extra structure and are called Gabor bimodules. We formulate a duality principle for standard module frames for Gabor bimodules which reduces to the well-known Gabor duality principle for twisted group $C^*$-algebras of a lattice in phase space. We lift all these results to the matrix algebra level an…
We perform a time-frequency analysis of Fourier multipliers and, more generally, pseudodifferential operators with symbols of Gevrey, analytic and ultra-analytic regularity. As an application we show that Gabor frames, which provide optimally sparse decompositions for Schroedinger-type propagators, reveal to be an even more efficient tool for representing solutions to a wide class of evolution operators with constant coefficients, including weakly hyperbolic and parabolic-type operators. Besides the class of operators, the main novelty of the paper is the proof of super-exponential (as opposit…
Let $\mathbb{H}$ be a separable Hilbert space. In this paper we establish a generalization of Walnut's representation and Janssen's representation of the $\mathbb{H}-$valued Gabor frame operator on $\mathbb{H}-$valued weighted amalgam spaces $W_{\mathbb{H}}(L^p,L^q_v)$, $1 \leq p, q \leq \infty$. Also we show that the frame operator is invertible on $W_{\mathbb{H}}(L^p,L^q_v)$, $1 \leq p, q \leq \infty$, if the window function is in the Wiener amalgam space $W_{\mathbb{H}}(L^{\infty},L^1_w)$. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to …
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…
The duality principle for group representations developed in \cite{DHL-JFA, HL_BLM} exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: The Wexler-Raz biorthogonality and the Fundamental Identity of Gabor analysis. In this paper we will show that these fundamental properties remain to be true for general projective unitary group representations. The main purpose of this paper is present a m…
This paper addresses the natural question: ``How should frames be compared?'' We answer this question by quantifying the overcompleteness of all frames with the same index set. We introduce the concept of a frame measure function: a function which maps each frame to a continuous function. The comparison of these functions induces an equivalence and partial order that allows for a meaningful comparison of frames indexed by the same set. We define the ultrafilter measure function, an explicit frame measure function that we show is contained both algebraically and topologically inside all frame m…
Motivated by potential applications in multiplexing and by recent results on Gabor analysis with Hermite windows due to Gröchenig and Lyubarskii, we investigate vector-valued wavelet transforms and vector-valued wavelet frames, which constitute special cases of super-wavelets, with a particular attention to the case when the analyzing wavelet vector is related to Fourier transforms of Laguerre functions. We construct an isometric isomorphism between $L^{2}(\mathbb{R}^{+},\mathbf{C}^{n})$ and poly-Bergman spaces, with a view to relate the sampling sequences in the poly-Bergman spaces to the wav…
Motivated by the recent work of Bownik and Ross \cite{BR}, and Jakobsen and Lemvig \cite{JL}, this article generalizes latest results on reproducing formulas for generalized translation invariant (GTI) systems to the setting of super-spaces over a second countable locally compact abelian (LCA) group $G$. To do so, we introduce the notion of a super-GTI system with finite sequences as generators from a super-space $L^2(G) \oplus \cdots \oplus L^2(G) $ ($N$ summands). We characterize the generators of two super-GTI systems in the super-space such that they form a super-dual frame pair. For this,…
