Related papers: Multi-window Gabor systems on discrete periodic se…

Gabor frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of engineering and science. Finding general and verifiable conditions which imply…

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…

G-frames are generalized frames which include ordinary frames, bounded invertible linear operators, as well as many recent generalizations of frames, e.g., bounded quasi-projectors and frames of subspaces. G-frames are natural…

We consider Gabor frames $\{e^{2\pi i bm \cdot} g(\cdot-ak)\}_{m,k \in \mathbb{Z}}$ 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…

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…

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 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…

Let $A$ be a finite subset of $L^2(\mathbb{R})$ and $p,q\in\mathbb{N}$. We characterize the Schauder basis properties in $L^2(\mathbb{R})$ of the Gabor system $$G(1,p/q,A)=\{e^{2\pi i m x}g(x-np/q) : m,n\in \mathbb{Z}, g\in A\},$$ with a…

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…

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…

In this work, we analyze Gabor frames for the Weyl--Heisenberg group and wavelet frames for the extended affine group. Firstly, we give necessary and sufficient conditions for the existence of nonstationary frames of translates. Using these…

We consider the problem in determining the countable sets $\Lambda$ in the time-frequency plane such that the Gabor system generated by the time-frequency shifts of the window $\chi_{[0,1]^d}$ associated with $\Lambda$ forms a Gabor…

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…

In this work we study families of pairs of window functions and lattices which lead to Gabor frames which all possess the same frame bounds. To be more precise, for every generalized Gaussian $g$, we will construct an uncountable family of…

In this paper Gabor system of certain type based on the unitary dual of the Heisenberg group $\mathbb{H}^n$ is introduced and a sufficient condition is obtained for the Gabor system to be a Bessel sequence for…

We find sufficient conditions on a compactly supported function $g$, $\supp g = [a,b]$ which guarantee that the Gabor system $$\mathcal{G}(g;\alpha,\beta)=\{e^{2\pi i \beta m x}g(x-\alpha n)\}_{m,n\in\mathbb{Z}}$$ is a frame for all $\alpha…

We prove that an overcomplete Gabor frame in $ \ell^2(\mathbf Z)$ by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in…

The concept of R-duals of a frame was introduced by Casazza, Kutyniok and Lammers in 2004, with the motivation to obtain a general version of the duality principle in Gabor analysis. For tight Gabor frames and Gabor Riesz bases the three…

A discrete frame for $L^2({\mathbb R}^d)$ is a countable sequence $\{e_j\}_{j\in J}$ in $L^2({\mathbb R}^d)$ together with real constants $0<A\leq B< \infty$ such that $$ A\|f\|_2^2 \leq \sum_{j\in J}|\langle f,e_j \rangle |^2 \leq…

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.…