We consider Gabor frames $\{e^{2πi bm \cdot} g(\cdot-ak)\}_{m,k \in \mathbb{Z}}$ with translation parameter $a=L/2$, modulation parameter $b \in (0,2/L)$ and a window function $g \in C^n(\mathbb{R})$ supported on $[x_0,x_0+L]$ and non-zero on $(x_0,x_0+L)$ for $L>0$ and $x_0\in \mathbb{R}$. The set of all dual windows $h \in L^2(\mathbb{R})$ with sufficiently small support is parametrized by $1$-periodic measurable functions $z$. Each dual window $h$ is given explicitly in terms of the function $z$ in such a way that desirable properties (e.g., symmetry, boundedness and smoothness) of $h$ are directly linked to $z$. We derive easily verifiable conditions on the function $z$ that guarantee, in fact, characterize, compactly supported dual windows $h$ with the same smoothness, i.e., $h \in C^n(\mathbb{R})$. The construction of dual windows is valid for all values of the smoothness index $n \in \mathbb{Z}_{\ge 0} \cup \{\infty\}$ and for all values of the modulation parameter $b<2/L$; since $a=L/2$, this allows for arbitrarily small redundancy $(ab)^{-1}>1$. We show that the smoothness of $h$ is optimal, i.e., if $g \notin C^{n+1}(\mathbb{R})$ then, in general, a dual window $h$ in $C^{n+1}(\mathbb{R})$ does not exist.
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 identify a class of continuous compactly supported functions for which the known part of the Gabor frame set can be extended. At least for functions with support on an interval of length two, the curve determining the set touches the known obstructions. Easy verifiable sufficient conditions for a function to belong to the class are derived, and it is shown that the B-splines $B_N, N\ge 2,$ as well as certain "continuous and truncated" versions of several classical functions (e.g., the Gaussian and the two-sided exponential function) belong to the class. The sufficient conditions for the fra…
The geometry of fundamental domains of lattices was used by Han and Wang to construct multivariate Gabor frames for separable lattices. We build upon their results to obtain Gabor frames with smooth and compactly supported window functions. For this purpose we study pairs of lattices which have equal density and allow for a common compact and star-shaped fundamental domain. The results are then extended to a larger class of lattices via symplectic equivalence.
This paper examines linear independence of shearlet systems. This property has already been studied for wavelets and other systems such as, for instance, for Gabor systems. In fact, for Gabor systems this problem is commonly known as the HRT conjecture. In this paper we present a proof of linear independence of compactly supported separable shearlet systems. For this, we employ a sampling strategy to utilize the structure of an implicitly given underlying oversampled wavelet system as well as the shape of the supports of the shearlet elements.
We investigate the structural properties of dual systems for nonstationary Gabor frames. In particular, we prove that some inverse nonstationary Gabor frame operators admit a Walnut-like representation, i.e. the operator acting on a function can be described by weighted translates of that function, even when the original frame operator is not diagonal. In this case, which only occurs when compactly supported window functions are used, the canonical dual frame partially inherits the structure of the original frame, with differences that we describe in detail. Moreover, we determine a sufficient…
Nonstationary Gabor frames were recently introduced in adaptive signal analysis. They represent a natural generalization of classical Gabor frames by allowing for adaptivity of windows and lattice in either time or frequency. In this paper we show a general existence result for this family of frames. We then give a perturbation result for nonstationary Gabor frames and construct nonstationary Gabor frames with non-compactly supported windows from a related painless nonorthogonal expansion. Finally, the theoretical results are illustrated by two examples of practical relevance.
In this paper we investigate the computational aspects of some recently proposed iterative methods for approximating the canonical tight and canonical dual window of a Gabor frame (g,a,b). The iterations start with the window g while the iteration steps comprise the window g, the k^{th} iterand γ_{k}, the frame operators S and S_{k} corresponding to (g,a,b) and (γ_{k},a,b), respectively, and a number of scalars. The structure of the iteration step of the method is determined by the envisaged convergence order m of the method. We consider two strategies for scaling the terms in the iteration st…
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 fu…
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…
Nonstationary Gabor frames, recently introduced in adaptive signal analysis, represent a natural generalization of classical Gabor frames by allowing for adaptivity of windows and lattice in either time or frequency. Due to the lack of a complete lattice structure, perfect reconstruction is in general not feasible from coefficients obtained from nonstationary Gabor frames. In this paper it is shown that for nonstationary Gabor frames that are related to some known frames for which dual frames can be computed, good approximate reconstruction can be achieved by resorting to approximately dual fr…
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…
Partition of unities appear in many places in analysis. Typically they are generated by compactly supported functions with a certain regularity. In this paper we consider partition of unities obtained as integer-translates of entire functions restricted to finite intervals. We characterize the entire functions that lead to a partition of unity in this way, and we provide characterizations of the "cut-off" entire functions, considered as functions of a real variable, to have desired regularity. In particular we obtain partition of unities generated by functions with small support and desired re…
We show that multi-window Gabor frames with windows in the Wiener algebra $W(L^{\infty}, \ell^{1})$ are Banach frames for all Wiener amalgam spaces. As a byproduct of our results we positively answer an open question that was posed by [Krishtal and Okoudjou, Invertibility of the Gabor frame operator on the Wiener amalgam space, J. Approx. Theory, 153(2), 2008] and concerns the continuity of the canonical dual of a Gabor frame with a continuous generator in the Wiener algebra. The proofs are based on a recent version of Wiener's $1/f$ lemma.
We report on initial findings on Gabor systems with multivariate Gaussian window. Unlike the existing characterisation for dimension one in terms of lattice density, our results indicate that the behavior of Gaussians in higher-dimensional Gabor systems is intricate and further exploration is a valuable and challenging task.
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, …
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.
We prove that Gabor systems generated by certain scaled B-splines can be considered as perturbations of the Gabor systems generated by the Gaussian, with a deviation within an arbitrary small tolerance whenever the order $N$ of the B-spline is sufficiently large. As a consequence we show that for any choice of translation/modulation parameters $a,b>0$ with $ab<1,$ the scaled version of $B_N$ generates Gabor frames for $N$ sufficiently large. Considering the Gabor frame decomposition generated by the Gaussian and a dual window, the results lead to estimates of the deviation from perfect reconst…
In the present investigation we are linking noncommutative geometry over noncommutative tori with Gabor analysis, where the first has its roots in operator algebras and the second in time-frequency analysis. Therefore we are in the position to invoke modern methods of operator algebras, e.g. topological stable rank of Banach algebras, to exploit the deeper properties of Gabor frames. Furthermore we are able to extend results due to Connes and Rieffel on projective modules over noncommutative tori to Banach algebras, which arise in a natural manner in Gabor analysis. The main goal of this inves…
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…
Let $H$ be a locally compact group, $K$ be an LCA group, $τ:H\to Aut(K)$ be a continuous homomorphism and $G_τ=H\ltimes_τK$ be the semi-direct product of $H$ and $K$ with respect to the continuous homomorphism $τ$. In this article we introduce the $τ\times\hatτ$-time frequency group $G_{τ\times\hatτ}$. We define the $τ\times\hatτ$-continuous Gabor transform of $f\in L^2(G_τ)$ with respect to a window function $u\in L^2(K)$ as a function defined on $G_{τ\times\hatτ}$. It is also shown that the $τ\times\hatτ$-continuous Gabor transform satisfies the Plancherel Theorem and reconstruction formula.…
In a recent paper in Appl. Comput. Harmon. Anal. 38(2), 196--221 (2014) we have introduced and studied the notion of weak Hamiltonian deformation of a Gabor (=Weyl-Heisenberg) frame. In this Note we use these results to prove that one can modify the window of a Gabor frame using certain metaplectic operators provided that one modifies only a finite number of points of the frame lattice.
We obtain characterizations of asymptotic properties of Schwartz distribution by using Gabor frames. Our characterizations are indeed Tauberian theorems for shift asymptotics (S-asymptotics) in terms of short-time Fourier transforms with respect to windows generating Gabor frames. For it, we show that the Gabor coefficient operator provides (topological) isomorphisms of the spaces of tempered distributions $\mathcal{S}'(\mathbb{R}^d)$ and distributions of exponential type $\mathcal{K}'_{1}(\mathbb{R}^{d})$ onto their images.
We consider the problem in determining the countable sets $Λ$ in the time-frequency plane such that the Gabor system generated by the time-frequency shifts of the window $χ_{[0,1]^d}$ associated with $Λ$ forms a Gabor orthonormal basis for $ L^2({\Bbb R}^d)$. We show that, if this is the case, the translates by elements $Λ$ of the unit cube in ${\Bbb R}^{2d}$ must tile the time-frequency space ${\Bbb R}^{2d}$. By studying the possible structure of such tiling sets, we completely classify all such admissible sets $Λ$ of time-frequency shifts when $d=1,2$. Moreover, an inductive procedure for co…
We prove stability results for a class of Gabor frames in $ L^2(\R)$. We consider window functions in the Sobolev spaces $H^1_0(\R)$ and B-splines of order $p\ge 1$. Our results can be used to describe the effect of the timing jitters in the $p$-order hold models of signal reconstruction.
We establish novel uniqueness results for the Gabor phase retrieval problem: if $\mathcal{G} : L^2(\mathbb{R}) \to L^2(\mathbb{R}^2)$ denotes the Gabor transform then every $f \in L^4[-\tfrac{c}{2},\tfrac{c}{2}]$ is determined up to a global phase by the values $|\mathcal{G}f(x,ω)|$ where $(x,ω)$ are points on the lattice $b^{-1}\mathbb{Z} \times (2c)^{-1}\mathbb{Z}$ and $b>0$ is an arbitrary positive constant. This for the first time shows that compactly-supported, complex-valued functions can be uniquely reconstructed from lattice samples of their spectrogram. Moreover, by making use of rece…
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.
We develop a theory of discrete directional Gabor frames for functions defined on the $d$-dimensional Euclidean space. Our construction incorporates the concept of ridge functions into the theory of isotropic Gabor systems, in order to develop an anisotropic Gabor system with strong directional sensitivity. We present sufficient conditions on a window function $g$ and a sampling set $Λ$ for the corresponding directional Gabor system $\{g_{m,t,u}\}_{(m,t,u)\inΛ}$ to form a discrete frame. Explicit estimates on the frame bounds are developed. A numerical implementation of our scheme is also pres…
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…
Gabor frames play a vital role not only modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this work we present a non-trivial generalization of Gabor frames to the quaternionic case and give new density results. The key tool is the two-sided windowed quaternionic Fourier transform (WQFT). As in the complex case, we want to write the WQFT as an inner product between a quaternion-valued signal and shifts and modulates of a real-valued window function. We demonstrate a Heisenberg uncertainty principle and for the…
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.
We study the fractal uncertainty principle in the joint time-frequency representation, and we prove a version for the Short-Time Fourier transform with Gaussian window on the modulation spaces. This can equivalently be formulated in terms of projection operators on the Bargmann-Fock spaces of entire functions. Specifically for signals in $L^2(\mathbb{R}^d)$, we obtain norm estimates of Daubechies' time-frequency localization operator localizing on porous sets. The proof is based on the maximal Nyquist density of such sets, and for multidimensional Cantor iterates we derive explicit upper bound…
We study sharp frame bounds of Gabor frames with the standard Gaussian window and prove that the square lattice optimizes both the lower and the upper frame bound among all rectangular lattices. This proves a conjecture of Floch, Alard & Berrou (as reformulated by Strohmer & Beaver). The proof is based on refined log-convexity/concavity estimates for the Jacobi theta functions $θ_3$ and $θ_4$.
Let $g$ be a totally positive function of finite type. Then the Gabor set $\{e^{2πi βl t} g(t-αk), k,l \in Z \}$ is a frame for $L^2(R)$, if and only if $αβ<1$. This result is a first positive contribution to a conjecture of I.\ Daubechies from 1990. So far the complete characterization of lattice parameters $α, β$ that generate a frame has been known for only six window functions $g$. Our main result now provides an uncountable class of functions. As a byproduct of the proof method we derive new sampling theorems in shift-invariant spaces and obtain the correct Nyquist rate.
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 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.
The time-frequency content of a signal can be measured by the Gabor transform or windowed Fourier transform. This is a function defined on phase space that is computed by taking the Fourier transform of the product of the signal against a translate of a fixed window. The problem of finding signals with Gabor transform that are maximally concentrated within a given region of phase space is discussed. It is well known that such problems give rise to an eigenvalue problem for an associated self-adjoint, positive {\em concentration} operator that has its spectrum contained in the unit interval. In…
We consider Gabor Riesz sequences generated by a lattice $Λ\subset \mathbb{R}^2$ and a window function $g \in L^2(\mathbb{R})$ which is well localized in both time and frequency. When $g$ belongs to the Feichtinger algebra, we prove that only those time-frequency shifts with parameters from the lattice $Λ$ leave the corresponding Gabor space invariant. This improves on earlier results where only lattices of rational density were considered. A slightly weaker result is proved - again for lattices of general density - under the regularity assumptions of the classical Balian-Low theorem, where bo…
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…
