Density, Overcompleteness, and Localization of Frames. II. Gabor Systems

This work develops a quantitative framework for describing the overcompleteness of a large class of frames. A previous article introduced 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. This article shows that those abstract results yield an array of new implications for irregular Gabor frames. Additionally, various Nyquist density results for Gabor frames are recovered as special cases, and in the process both their meaning and implications are clarified. New results are obtained on the excess and overcompleteness of Gabor frames, on the relationship between frame bounds and density, and on the structure of the dual frame of an irregular Gabor frame. More generally, these results apply both to Gabor frames and to systems of Gabor molecules, whose elements share only a common envelope of concentration in the time-frequency plane. The notions of localization and related approximation properties are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. In this article, a comprehensive examination of the interrelations among these localization and approximation concepts is made, with most implications shown to be sharp.     

Excess of a class of g-frames

In this paper, we study the relationship between frames for the super Hilbert space H⊕H and g-frames for H with respect to C². We show that a g-frame associated with a frame for H⊕H remains a g-frame whenever any one of its elements is removed. Furthermore, we show that the excess of such a g-frame is at least dimH.     

Perturbations of Banach Frames and Atomic Decompositions

Banach frames and atomic decompositions are sequences that have basis-like properties but which need not be bases. In particular, they allow elements of a Banach space to be written as linear combinations of the frame or atomic decomposition elements in a stable manner. In this paper we prove several functional — analytic properties of these decompositions, and show how these properties apply to Gabor and wavelet systems. We first prove that frames and atomic decompositions are stable under small perturbations. This is inspired by corresponding classical perturbation results for bases, including the Paley — Wiener basis stability criteria and the perturbation theorem el kato. We introduce new and weaker conditions which ensure the desired stability. We then prove quality properties of atomic decompositions and consider some consequences for Hilbert frames. Finally, we demonstrate how our results apply in the practical case of Gabor systems in weighted L² spaces. Such systems can form atomic decompositions for L²_w(IR), but cannot form Hilbert frames but L²_w(IR) unless the weight is trivial.     

Continuous Frames, Function Spaces, and the Discretization Problem

A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. Associated to a given continuous frame we construct certain Banach spaces. Many classical function spaces can be identified as such spaces. We provide a general method to derive Banach frames and atomic decompositions for these Banach spaces by sampling the continuous frame. This is done by generalizing the coorbit space theory developed by Feichtinger and Gröchenig. As an important tool the concept of localization of frames is extended to continuous frames. As a byproduct we give a partial answer to the question raised by Ali, Antoine, and Gazeau whether any continuous frame admits a corresponding discrete realization generated by sampling.     

Random Tight Frames

We introduce probabilistic frames to study finite frames whose elements are chosen at random. While finite tight frames generalize orthonormal bases by allowing redundancy, independent, uniformly distributed points on the sphere approximately form a finite unit norm tight frame (FUNTF). In the present paper, we develop probabilistic versions of tight frames and FUNTFs to significantly weaken the requirements on the random choice of points to obtain an approximate finite tight frame. Namely, points can be chosen from any probabilistic tight frame, they do not have to be identically distributed, nor have unit norm. We also observe that classes of random matrices used in compressed sensing are induced by probabilistic tight frames.     

An uncertainty principle for finite frames

We consider the notion of uncertainty for finite frames. Using a difference operator inspired by the Gauss-Hermite differential equation we obtain a time-frequency measure for finite frames. We then find the minimizer of the measure over all equal norm Parseval frames, dependent on the dimension of the space and the number of elements in the frame. Next we show that given a frame one can find the dual frame that minimizes this time-frequency measure, generalizing some work of Daubechies, Landau and Landau to the finite case and extending some recent work on Sobolev duals for finite frames.     

A Characterization of Schauder Frames Which Are Near-Schauder Bases

A basic problem of interest in connection with the study of Schauder frames in Banach spaces is that of characterizing those Schauder frames which can essentially be regarded as Schauder bases. In this paper, we give a solution to this problem using the notion of the minimal-associated sequence spaces and the minimal-associated reconstruction operators for Schauder frames. We prove that a Schauder frame is a near-Schauder basis if and only if the kernel of the minimal-associated reconstruction operator contains no copy of c ₀. In particular, a Schauder frame of a Banach space with no copy of c ₀ is a near-Schauder basis if and only if the minimal-associated sequence space contains no copy of c ₀. In these cases, the minimal-associated reconstruction operator has a finite dimensional kernel and the dimension of the kernel is exactly the excess of the near-Schauder basis. Using these results, we make related applications on Besselian frames and near-Riesz bases.     

Dualizable Shearlet Frames and Sparse Approximation

Shearlet systems have been introduced as directional representation systems, which provide optimally sparse approximations of a certain model class of functions governed by anisotropic features while allowing faithful numerical realizations by a unified treatment of the continuum and digital realm. They are redundant systems, and their frame properties have been extensively studied. In contrast to certain band-limited shearlets, compactly supported shearlets provide high spatial localization but do not constitute Parseval frames. Thus reconstruction of a signal from shearlet coefficients requires knowledge of a dual frame. However, no closed and easily computable form of any dual frame is known. In this paper, we introduce the class of dualizable shearlet systems, which consist of compactly supported elements and can be proved to form frames for \(L^2({\mathbb {R}}^2)\). For each such dualizable shearlet system, we then provide an explicit construction of an associated dual frame, which can be stated in closed form and is efficiently computed. We also show that dualizable shearlet frames still provide near optimal sparse approximations of anisotropic features.     

Representation of the inverse of a frame multiplier

Certain mathematical objects appear in a lot of scientific disciplines, like physics, signal processing and, naturally, mathematics. In a general setting they can be described as frame multipliers, consisting of analysis, multiplication by a fixed sequence (called the symbol), and synthesis. In this paper we show a surprising result about the inverse of such operators, if any, as well as new results about a core concept of frame theory, dual frames. We show that for semi-normalized symbols, the inverse of any invertible frame multiplier can always be represented as a frame multiplier with the reciprocal symbol and dual frames of the given ones. Furthermore, one of those dual frames is uniquely determined and the other one can be arbitrarily chosen. We investigate sufficient conditions for the special case, when both dual frames can be chosen to be the canonical duals. In connection to the above, we show that the set of dual frames determines a frame uniquely. Furthermore, for a given frame, the union of all coefficients of its dual frames is dense in ?²${?}^2$. We also introduce a class of frames (called pseudo-coherent frames), which includes Gabor frames and coherent frames, and investigate invertible pseudo-coherent frame multipliers, allowing a classification for frame-type operators for these frames. Finally, we give a numerical example for the invertibility of multipliers in the Gabor case.     

(p,Y)-算子框架的稳定性

In this paper we study the stability of（p,Y）-operator frames.We firstly discuss the relations between p-Bessel sequences（or p-frames） and（p,Y）-operator Bessel sequences（or（p,Y）-operator frames）.Through defining a new union,we prove that adding some elements to a given（p,Y）-operator frame,the resulted sequence will be still a（p,Y）-operator frame.We obtain a necessary and sufficient condition for a sequence of compound operators to be a（p,Y）operator frame.Lastly,we show that（p,Y）-operator frames for X are stable under some small perturbations.