Related papers: Gabor orthonormal bases generated by the unit cube…
We study uncertainty principles for orthonormal bases and sequences in $L^2(\R^d)$. As in the classical Heisenberg inequality we focus on the product of the dispersions of a function and its Fourier transform. In particular we prove that…
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…
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 apply the orthonormalization procedure previously introduced by two of us and adopted in connection with coherent states to Gabor frames and other examples. For instance, for Gabor frames we show how to construct $g(x)\in L^2(\Bbb{R})$…
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…
Motivated by a recent generalization of the Balian-Low theorem and by new research in wireless communications we analyze the construction of Wilson bases for general time-frequency lattices. We show that orthonormal Wilson bases for $\LtR$…
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…
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…
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…
We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ must be weakly periodic (the disjoint union…
Collections of time- and frequency-shifts of suitably chosen generators (Alltop or random vectors) proved successful for many applications in sparse recovery and related fields. It was shown in \cite{xia2005achieving} that taking a…
In this paper, we study multiwindow discrete Gabor $(M-D-G)$ systems $\mathcal{G}(g,L,M,N)$ on discrete periodic sets $\mathbb{S}$ and give some necessary and/or sufficient matrix-conditions for a $M-D-G$ system in $\ell^2(\mathbb{S})$ to…
Warped time-frequency systems have recently been introduced as a class of structured continuous frames for functions on the real line. Herein, we generalize this framework to the setting of functions of arbitrary dimensionality. After…
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…
We consider the cubic nonlinear Schr\"odinger equation posed on the spatial domain $\mathbb{R}\times \mathbb{T}^d$. We prove modified scattering and construct modified wave operators for small initial and final data respectively ($1\leq…
We show the existence of a family of frames of $L^2(\mathbb{R})$ which depend on a parameter $\alpha\in [0,1]$. If $\alpha=0$, we recover the usual Gabor frame, if $\alpha=1$ we obtain a frame system which is closely related to the so…
In the design and analysis of composite materials based on periodic arrangements of sub-units it is of paramount importance to control the emergent material symmetry in relation to the elastic response. The target material symmetry plays…
Motivated by problems on Brownian motion, we introduce a recursive scheme for a basis construction in the Hilbert space L^2(0,1) which is analogous to that of Haar and Walsh. More generally, we find a new decomposition theory for the…
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…
Let $\mathcal G\subset L^2(\mathbb R)$ be the subspace spanned by a Gabor Riesz sequence $(g,\Lambda)$ with $g\in L^2(\mathbb R)$ and a lattice $\Lambda\subset\mathbb R^2$ of rational density. It was shown recently that if $g$ is…