Pseudoframes for subspaces have been recently introduced by S. Li and H. Ogawa as a tool to analyze lower dimensional data with arbitrary flexibility of both the analyzing and the dual sequence. In this paper we study Gabor pseudoframes for affine subspaces by focusing on geometrical properties of their associated sets of parameters. We first introduce a new notion of Beurling dimension for dis...

This paper introduces the concept of Bessel multipliers. These operators are defined by a fixed multiplication pattern, which is inserted between the analysis and synthesis operators. The proposed concept unifies the approach used for Gabor multipliers for arbitrary analysis/synthesis systems, which form Bessel sequences, like wavelet or irregular Gabor frames. The basic properties of this clas...

This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames F = {f i } i∈I and E = {e j } j∈G (G a discrete abelian group), relating the decay of the expansion of the elements of F in terms of the elements of E via a map a : I → G. A fundamental set of equalities are shown ...

Frames have applications in numerous fields of mathematics and engineering. The fundamental property of frames which makes them so useful is their overcompleteness. In most applications, it is this overcompleteness that is exploited to yield a decomposition that is more stable, more robust, or more compact than is possible using nonredundant systems. This work presents a quantitative framework ...

Classical and recent results on uncertainty principles for functions on finite Abelian groups relate the cardinality of the support of a function to the cardinality of the support of its Fourier transforms. We obtain corresponding results relating the support sizes of functions and their short–time Fourier transforms. We use our findings to construct a class of equal norm tight Gabor frames tha...

We consider three problems for Gabor frames that have recently received much attention. The first problem concerns the approximation of dual Gabor frames in L2(R) by finite-dimensional methods. Utilizing Wexler-Raz type duality relations we derive a method to approximate the dual Gabor frame, that is much simpler than previously proposed techniques. Furthermore it enables us to give estimates f...

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 ĝ(ξ) = ∏n j=1(1 + 2πiδjξ) −1 e 2 for δ1, . . . , δn ∈ R, c > 0 (in which case g is called totally positive of Gaussian type). In analogy to Beurling’s sampling theorem for the Paley-Wiener space of entire functions...

Let g ∈ L2(R) be a compactly supported function, whose integertranslates {Tkg}k∈Z form a partition of unity. We prove that for certain translationand modulation parameters, such a function g generates a Gabor frame, with a (non-canonical) dual generated by a finite linear combination h of the functions {Tkg}k∈Z; the coefficients in the linear combination are given explicitly. Thus, h has compac...

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) ∈ L2(R) in such a way the functions gn(x) = e 1g(x+an2), n ∈ Z2 and a some positive real number, are mutually orthogonal. We discuss in some details the role of the lattice nat...

We introduce a new notion for the deformation of Gabor systems. Such deformations are in general nonlinear and, in particular, include the standard jitter error and linear deformations of phase space. With this new notion we prove a strong deformation result for Gabor frames and Gabor Riesz sequences that covers the known perturbation and deformation results. Our proof of the deformation theore...