For a class of compactly supported windows we characterize the frame property for a Gabor system $\mts,$ for translation parameters $a$ belonging to a certain range depending on the support size. We show that the obstructions to the frame property are located on a countable number of "curves." For functions that are positive on the interior of the support these obstructions do not appear, and the considered region in the $(a,b)$ plane is fully contained in the frame set. In particular this confirms a recent conjecture about B-splines by Gröchenig in that particular region. We prove that the full conjecture is true if it can be proved in a certain "hyperbolic strip."
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.
We show the full structure of the frame set for the Gabor system $\mathcal{G}(g;α,β):=\{e^{-2πi mβ\cdot}g(\cdot-nα):m,n\in\Bbb Z\}$ with the window being the Haar function $g=-χ_{[-1/2,0)}+χ_{[0,1/2)}$. The strategy of this paper is to introduce the piecewise linear transformation $\mathcal{M}$ on the unit circle, and to provide a complete characterization of structures for its (symmetric) maximal invariant sets. This transformation is related to the famous three gap theorem of Steinhaus which may be of independent interest. Furthermore, a classical criterion on Gabor frames is improved, which…
A Gabor system generated by a window function $g\in L^2(\mathbb{R}^d)$ and a separable set $Λ\times Γ\subset \mathbb{R}^{2d}$ is the collection of time-frequency shifts of $g$ given by $\mathcal G(g, Λ\times Γ) = \left\{ e^{2πi ξ\cdot t}g(t-x)\right\}_{ (x,ξ)\in Λ\times Γ}$. One of the fundamental problems in Gabor analysis is to characterize all windows and time-frequency sets that generate a Gabor frame or Gabor orthonormal basis. The case of Gabor orthonormal bases generated by characteristic functions $g=χ_Ω$ has been solved by Han and Wang. In this paper, we build on these results and obt…
Gabor frames have interested many mathematicians and physicists due to their potential applications in time-frequency analysis, in particular, signal processing. A Gabor system is a collection of vectors which is obtained by applying modulation and shift operators to non-zero functions in signal spaces. In many applications, for example, signal processing related to Gabor systems, the corresponding shifts may not be uniform. That is, the set associated with shifts may not be a group under usual addition. We analyze discrete vector-valued nonuniform Gabor frames (DVNUG frames, in short) in disc…
我们证明了窗口在Wiener代数$W(L^{\infty}, \ell^{1})$中的多窗Gabor框架是所有Wiener汞齐空间的Banach框架。作为我们的结果的副产品，我们肯定地回答了由[Krishtal和Okoudjou，Invertibility of the Gabor frame operator on the Wiener amalgam space，J. Approx. Theory，153（2），2008]，并且涉及具有Wiener代数中的连续生成元的Gabor框架的典型对偶的连续性。证明是基于维纳的$1/f$引理的最新版本。
We consider Gabor frames generated by a general lattice and a window function that belongs to one of the following spaces: the Sobolev space $V_1 = H^1(\mathbb R^d)$, the weighted $L^2$-space $V_2 = L_{1 + |x|}^2(\mathbb R^d)$, and the space $V_3 = \mathbb H^1(\mathbb R^d) = V_1 \cap V_2$ consisting of all functions with finite uncertainty product; all these spaces can be described as modulation spaces with respect to suitable weighted $L^2$ spaces. In all cases, we prove that the space of Bessel vectors in $V_j$ is mapped bijectively onto itself by the Gabor frame operator. As a consequence, …
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 …
We study the frame properties of the Gabor systems $$\mathfrak{G}(g;α,β):=\{e^{2πi βm x}g(x-αn)\}_{m,n\in\mathbb{Z}}.$$ In particular, we prove that for Herglotz windows $g$ such systems always form a frame for $L^2(\mathbb{R})$ if $α,β>0$, $αβ\leq1$. For general rational windows $g\in L^2(\mathbb{R})$ we prove that $\mathfrak{G}(g;α,β)$ is a frame for $L^2(\mathbb{R})$ if $0<α,β$, $αβ<1$, $αβ\not\in\mathbb{Q}$ and $\hat{g}(ξ)\neq0$, $ξ>0$, thus confirming Daubechies conjecture for this class of functions. We also discuss some related questions, in particular sampling in shift-invariant subspa…
The frame set of a function $g\in L^2(\mathbb{R})$ is the subset of all parameters $(a, b)\in \mathbb{R}^2_+$ for which the time-frequency shifts of $g$ along $a\mathbb{Z}\times b\mathbb{Z}$ form a Gabor frame for $L^2(\mathbb{R}).$ In this paper, we investigate the frame set of a class of functions that we call \emph{generalized $B-$splines} and which includes the $B-$splines. In particular, we add many new points to the frame sets of these functions. In the process, we generalize and unify some recent results on the frame sets for this class of functions.
We obtain Gabor frame characterisations of modulation spaces defined via a class of translation-modulation invariant Banach spaces of distributions that was recently introduced in $[10]$. We show that these spaces admit an atomic decomposition through Gabor expansions and that they are characterised by summability properties of their Gabor coefficients. Furthermore, we construct a large space of admissible windows. This generalises several fundamental results for the classical modulation spaces $ M^{p,q}_{w}$. Due to the absence of solidity assumptions on the Banach spaces defining these modul…
In this paper, we show that the non-symmetric convex polytope cannot be served as a window function to produce a Gabor orthonormal basis by any time-frequency sets.
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 show that Hilbert-Schmidt operators can be used to define frame-like structures for $L^2(\mathbb{R}^d)$ over lattices in $\mathbb{R}^{2d}$ that include multi-window Gabor frames as a special case. These frame-like structures are called Gabor g-frames, as they are examples of g-frames as introduced by Sun. We show that Gabor g-frames share many properties of Gabor frames, including a Janssen representation and Wexler-Raz biorthogonality conditions. A central part of our analysis is a notion of Fourier series of periodic operators based on earlier work by Feichtinger and Kozek, where we show …
We consider sparseness properties of adaptive time-frequency representations obtained using nonstationary Gabor frames (NSGFs). NSGFs generalize classical Gabor frames by allowing for adaptivity in either time or frequency. It is known that the concept of painless nonorthogonal expansions generalizes to the nonstationary case, providing perfect reconstruction and an FFT based implementation for compactly supported window functions sampled at a certain density. It is also known that for some signal classes, NSGFs with flexible time resolution tend to provide sparser expansions than can be obtai…
We investigate the relation between two different mathematical problems: the construction of bounds on sphere packing density using Cohn-Elkies functions and the construction of Gabor frames for signal analysis. In particular, we present a general construction of Cohn-Elkies functions in arbitrary dimension, obtained from an approximate Wexel-Raz dual for Gabor frames with Gaussian window.
