Related papers: Characterization and computation of canonical tigh…
This survey offers a systematic and streamlined exposition of the most important characterizations of Gabor frames over a lattice. The goal is to collect the most important characterizations of Gabor frames and offer a systematic exposition…
Let $H$ be an infinite-dimensional separable Hilbert space and let $(X,d,\mu)$ be a metric measure space satisfying the doubling and upper Alhfors regularity conditions at small scale. We prove that every bounded continuous tight frame…
The topic of this paper are (multi-window) Gabor frames for signals over finite Abelian groups, generated by an arbitrary lattice within the finite time-frequency plane. Our generic approach covers simultaneously multi-dimensional signals…
In this paper we construct frames of Gabor type for the space $L^2_{rad}(\R^d)$ of radial $L^2$-functions, and more generally, for subspaces of modulation spaces consisting of radial distributions. Hereby, each frame element itself is a…
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…
We investigate Gabor frames on locally compact abelian groups with time-frequency shifts along non-separable, closed subgroups of the phase space. Density theorems in Gabor analysis state necessary conditions for a Gabor system to be a…
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…
Recent research has shown that the properties of overcomplete Gabor frames and frames arising from shift-invariant systems form a precise match with certain conditions that are necessary for a frame in $L^2(\mathbf R)$ to have a…
We study totally positive (TP) functions of finite type and exponential B-splines as window functions for Gabor frames. We establish the connection of the Zak transform of these two classes of functions and prove that the Zak transforms…
The frame set of a function $g\in L^2(\mathbb{R})$ is the set of all parameters $(a, b)\in \mathbb{R}^2_+$ for which the collection of time-frequency shifts of $g$ along $a\mathbb{Z}\times b\mathbb{Z}$ form a Gabor frame for…
We provide the foundations of a Hilbert space theory for the short-time Fourier transform (STFT) where the flat tori \begin{equation*} \mathbb{T}_{N}^2=\mathbb{R}^2/(\mathbb{Z}\times N\mathbb{Z})=[0,1]\times \lbrack 0,N] \end{equation*} act…
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…
We discuss the concepts of pseudo-dual frames and approximately dual frames, and illuminate their relationship to classical frames. Approximately dual frames are easier to construct than the classical dual frames, and might be tailored to…
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…
We study the existence of Gabor orthonormal bases with window the characteristic function of the set W=[0,a] U [b+a, b+1] of measure 1, with a, b>0. By the symmetries of the problem, we can restrict our attention to the case a<=1/2. We…
Given a window $\phi \in L^2(\mathbb R),$ and lattice parameters $\alpha, \beta>0,$ we introduce a bimodal Wilson system $\mathcal{W}(\phi, \alpha, \beta)$ consisting of linear combinations of at most two elements from an associated Gabor…
A Wilson system is a collection of finite linear combinations of time frequency shifts of a square integrable function. In this paper we use the fact that a Wilson system is a shift-invariant system to explore its relationship with Gabor…
Although frames, which are a generalization of bases, are important tools used in signal processing, their potential in other fields of engineering and applied mathematics (e.g. acoustics) has not been fully explored yet. Gabor frames, that…
Time-frequency analysis, such as the Gabor transform, plays an important role in many signal processing applications. The redundancy of such representations is often directly related to the computational load of any algorithm operating in…
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…