Related papers: Gabor frames for functions supported on a semi-axi…
Given a lattice $\Lambda$ in a locally compact abelian group $G$ and a measurable subset $\Omega$ with finite and positive measure, then the set of characters associated to the dual lattice form a frame for $L^2(\Omega)$ if and only if the…
In this paper a large class of universal windows for Gabor frames (Weyl-Heisenberg frames) is constructed. These windows have the fundamental property that every overcritical rectangular lattice generates a Gabor frame. Likewise, every…
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 $(g_{nm})_{n,m\in Z}$ 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…
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…
We study finite systems of vectors whose frame operator matrices are unitarily equivalent, via explicit and computationally efficient unitary transformations, to block-diagonal matrices. We call such systems block-equivalent. We show that a…
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 investigate sharp frame bounds of Gabor frames with chirped Gaussians and rectangular lattices or, equivalently, the case of the standard Gaussian and general lattices. We prove that for even redundancy and standard Gaussian window the…
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…
We consider the following equations: \begin{equation*} \left\{\begin{array}{ll} (-\triangle)^{\alpha/2}u(x)=f(v(x)), \\ (-\triangle)^{\beta/2}v(x)=g(u(x)), &x \in R^{n},\\ u,v\geq 0, &x \in R^{n}, \end{array} \right. \end{equation*} for…
We examine Fourier frames and, more generally, frame measures for different probability measures. We prove that if a measure has an associated frame measure, then it must have a certain uniformity in the sense that the weight is distributed…
Let $1\leq p\leq 2$ and let $\Lambda = \{\lambda_n\}_{n\in \mathbb{N}} \subseteq \mathbb{R}$ be an arbitrary subset. We prove that for any $g\in M^p(\mathbb{R})$ with $1\leq p\leq 2$ the system of translates $\{g(x-\lambda_n)\}_{n\in…
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…
The construction of finite tight Gabor frames plays an important role in many applications. These applications include significant ones in signal and image processing. We explore when constant amplitude zero autocorrelation (CAZAC)…
We study results related to a conjecture formulated by Strohmer and Beaver about optimal Gaussian Gabor frame set-ups. Our attention will be restricted to the case of Gabor systems with standard Gaussian window and rectangular lattices of…
In this paper we introduce and investigate the concept of reproducing pairs which generalizes continuous frames. We will introduce a concept that represents a unifying way to look at certain continuous frames (resp. reproducing pairs) on…
Let $G$ be a connected, simply connected, nilpotent Lie group whose irreducible unitary representations are square-integrable modulo the center. We obtain characterization results for reproducing formulas associated with the left…
We establish the existence of positive solutions for a system of coupled fourth-order partial differential equations on a bounded domain $\Omega \subset \mathbb{R}^n$\begin{align*} \left\{\begin{array}{l} \Delta^2u_1 +\beta_1 \Delta…
In this article we are going to discuss the conjecture of Strohmer and Beaver for Gaussian Gabor systems. It asks for an optimal sampling pattern in the time-frequency plane, where optimality is measured in terms of the condition number of…
We consider the question whether, given a countable system of lattices $(\Gamma_j)_{j \in J}$ in a locally compact abelian group $G$, there exists a sequence of functions $(g_j)_{j \in J}$ such that the resulting generalized shift-invariant…