Related papers: Irregular sampling for hyperbolic secant type func…

We show that $(g_2,a,b)$ is a Gabor frame when $a>0, b>0, ab<1$ and $g_2(t)=({1/2}\pi \gamma)^{{1/2}} (\cosh \pi \gamma t)^{-1}$ is a hyperbolic secant with scaling parameter $\gamma >0$. This is accomplished by expressing the Zak transform…

We study nonuniform sampling in shift-invariant spaces and the construction of Gabor frames with respect to the class of totally positive functions whose Fourier transform factors as $ \hat g(\xi)= \prod_{j=1}^n (1+2\pi i\delta_j\xi)^{-1}…

We introduce two families of generators (functions) $\mathcal{G}$ that consist of entire and meromorphic functions enjoying a certain periodicity property and contain the classical Gaussian and hyperbolic secant generators. Sharp results…

We study the frame properties of the Gabor systems $$\mathfrak{G}(g;\alpha,\beta):=\{e^{2\pi i \beta m x}g(x-\alpha n)\}_{m,n\in\mathbb{Z}}.$$ In particular, we prove that for Herglotz windows $g$ such systems always form a frame for…

Let $g\in L^2(\mathbb{R})$ be a strictly decreasing continuous function supported on $\mathbb{R}_+$ such that for all $t > 0$ we have $g(x+t)\le q(t)g(x)$ for some $q(t)<1$. We prove that the Gabor system…

We study the problem of sampling with derivatives in shift-invariant spaces generated by totally-positive functions of Gaussian type or by the hyperbolic secant. We provide sharp conditions in terms of weighted Beurling densities. As a…

A Gabor system generated by a window function $g\in L^2(\mathbb{R}^d)$ and a separable set $\Lambda\times \Gamma \subset \mathbb{R}^{2d}$ is the collection of time-frequency shifts of $g$ given by $\mathcal G(g, \Lambda\times \Gamma) =…

We consider the frame property of the Gabor system G(g, {\alpha}, {\beta}) = {e2{\pi}i{\beta}nt g(t - {\alpha}m) : m, n \in Z} for the case of rational oversampling, i.e. {\alpha}, {\beta} \in Q. A 'rational' analogue of the Ron-Shen…

We study the space spanned by the integer shifts of a bivariate Gaussian function and the problem of reconstructing any function in that space from samples scattered across the plane. We identify a large class of lattices, or more generally…

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 consider the following problem: given a set $\Lambda \subset \mathbb{R} \times \mathbb{R}$ and $p \neq 2$, does there exist a function $g \in L^p(\mathbb{R})$ such that the Gabor system $\{g(x-t) e^{2 \pi isx}\}$, $(t,s) \in \Lambda$,…

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 give a construction of Gabor type frames for suitable separable subspaces of the non-separable Hilbert spaces $AP_2({\mathbb R})$ of almost periodic functions of one variable. Furthermore we determine a non-countable generalized frame…

We show the full structure of the frame set for the Gabor system $\mathcal{G}(g;\alpha,\beta):=\{e^{-2\pi i m\beta\cdot}g(\cdot-n\alpha):m,n\in\Bbb Z\}$ with the window being the Haar function $g=-\chi_{[-1/2,0)}+\chi_{[0,1/2)}$. The…

For a window $g\in L^2(\mathbb{R})$, the subset of all lattice parameters $(a, b)\in \mathbb{R}^2_+$ such that $\mathcal{G}(g,a,b)=\{e^{2\pi ib m\cdot}g(\cdot-a k) : k, m\in\mathbb{Z}\}$ forms a frame for $L^2(\mathbb{R})$ is known as the…

In this article, we consider a variation of the existence of Gabor frames in a probabilistic setting, in which we consider time-frequency shifts taken over random-periodic sets. We demonstrate that the method of selecting random-periodic…

Let $g$ be a totally positive function of finite type. Then the Gabor set $\{e^{2\pi i \beta l t} g(t-\alpha k), k,l \in Z \}$ is a frame for $L^2(R)$, if and only if $\alpha \beta <1$. This result is a first positive contribution to a…

A Gabor system generated by a window function $\phi$ and a rectangular lattice $a \Z\times \Z/b$ is given by $${\mathcal G}(\phi, a \Z\times \Z/b):=\{e^{-2\pi i n t/b} \phi(t- m a):\ (m, n)\in \Z\times \Z\}.$$ One of fundamental problems in…

We study sharp frame bounds of Gabor systems over rectangular lattices for different windows and integer oversampling rate. In some cases we obtain optimality results for the square lattice, while in other cases the lattices optimizing the…

Let $(g_{nm})_{n,m\in Z}$ be a Gabor frame for $L_2(R)$ for given window $g$. We show that the window $h^0=S^{-1/2} g$ that generates the canonically associated tight Gabor frame minimizes $\|g-h\|$ among all windows $h$ generating a…