Rational time-frequency multi-window subspace Gabor frames and their Gabor duals

This paper addresses the theory of multi-window subspace Gabor frame with rational time-frequency parameter products.With the help of a suitable Zak transform matrix,we characterize multi-window subspace Gabor frames,Riesz bases,orthonormal bases and the uniqueness of Gabor duals of type I and type II.Using these characterizations we obtain a class of multi-window subspace Gabor frames,Riesz bases,orthonormal bases,and at the same time we derive an explicit expression of their Gabor duals of type I and type II.As an application of the above results,we obtain characterizations of multi-window Gabor frames,Riesz bases and orthonormal bases for L2(R),and derive a parametric expression of Gabor duals for multi-window Gabor frames in L2(R).     

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.     

APPROXIMATE DUALITY OF g-FRAMES IN HILBERT SPACES

In this article, we introduce and characterize approximate duality for g-frames. We get some important properties and applications of approximate duals. We also obtain some new results in approximate duality of frames, and generalize some of the known results in approximate duality of frames to g-frames. We also get some results for fusion frames, and perturbation of approximately dual g-frames. We show that approximate duals are stable under small perturbations and they are useful for erasures and reconstruction.     

Approximate duality of g-frames in Hilbert spaces

In this article, we introduce and characterize approximate duality for g-frames. We get some important properties and applications of approximate duals. We also obtain some new results in approximate duality of frames, and generalize some of the known results in approximate duality of frames to g-frames. We also get some results for fusion frames, and perturbation of approximately dual g-frames. We show that approximate duals are stable under small perturbations and they are useful for erasures and reconstruction.     

Time-domain characterization of multiwindow Gabor systems on discrete periodic sets

This paper addresses multiwindow Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. We give some necessary and/or sufficient conditions for multiwindow Gabor systems to foe frames on discrete periodic sets, and characterize two multiwindow Gabor Bessel sequences to foe dual frames on discrete periodic sets. For a given multiwindow Gabor frame, we derive all its Gabor duals, among which we obtain an explicit expression of the canonical Gabor dual. In addition, we generalize multiwindow Gabor systems to the case of a different sampling rate for each window, and investigate multiwindow Gabor frames and dual frames in this case. We also show the properties of the multiwindow Gabor systems are essentially not changed by replacing the exponential kernel with other kernels.     

Affine Density,Frame Bounds,and the Admissibility Condition for Wavelet Frames

For a large class of irregular wavelet frames we derive a fundamental relationship between the affine density of the set of indices, the frame bounds, and the admissibility constant of the wavelet. Several implications of this theorem are studied. For instance, this result reveals one reason why wavelet systems do not display a Nyquist phenomenon analogous to Gabor systems, a question asked in Daubechies' Ten Lectures book. It also implies that the affine density of the set of indices associated with a tight wavelet frame has to be uniform. Finally, we show that affine density conditions can even be used to characterize the existence of wavelet frames, thus serving, in particular, as sufficient conditions.     

Perturbation of wavelet and Gabor frames

In this work two aspects of theory of frames are presented: a side necessary condition on irregular wavelet frames is obtained, another perturbation of wavelet and Gabor frames is considered. Specifically,we present the results obtained on frame stability when one disturbs the mother of wavelet frame, or the parameter of dilatation, and in Gabor frames when the generating function or the parameter of translation are perturbed. In all cases we work without demanding compactness of the support, neither on the generating function, nor on its Fourier transform.     

Characterization of rationally sampled quaternionic dual Gabor frames

This paper addresses quaternionic dual Gabor frames under the condition that the products of time-frequency shift parameters are rational numbers. For a general overcomplete quaternionic Gabor frame with the product of time-frequency shift parameters not equal to $\frac{1}{2}$, we show that its corresponding frame and translation operators do not commute, which leads to its canonical dual frame not having the Gabor structure, but it may have other dual frames with Gabor structure. We characterize when two quaternionic Gabor Bessel sequences form a pair of dual frames, and present a class of quaternionic dual Gabor frames. We also characterize quaternionic Gabor Riesz bases and prove that their canonical dual frames have Gabor structure.     

Super Gabor frames on discrete periodic sets

Due to its potential applications in multiplexing techniques such as time division multiple access and frequency division multiple access, superframe has interested some mathematicians and engineering specialists. In this paper, we investigate super Gabor systems on discrete periodic sets in terms of a suitable Zak transform matrix, which can model signals to appear periodically but intermittently. Complete super Gabor systems, super Gabor frames and Gabor duals for super Gabor frames on discrete periodic sets are characterized; An explicit expression of Gabor duals is established, and the uniqueness of Gabor duals is characterized. On the other hand, discrete periodic sets admitting complete super Gabor systems, super Gabor frames, super Gabor Riesz bases are also characterized. Some examples are also provided to illustrate the general theory.     

Intrinsic Localization of Frames

Several concepts for the localization of a frame are studied. The intrinsic localization of a frame is defined by the decay properties of its Gramian matrix. Our main result asserts that the canonical dual frame possesses the same intrinsic localization as the original frame. The proof relies heavily on Banach algebra techniques, in particular on recent spectral invariance properties for certain Banach algebras of infinite matrices. Intrinsically localized frames extend in a natural way to Banach frames for a class of associated Banach spaces which are defined by weighted ℓ^p-coefficients of their frame expansions. As an example, the time--frequency concentration of distributions is characterized by means of localized (nonuniform) Gabor frames.     

On the duality of fusion frames

The fusion frames were considered recently by P.G. Casazza, G. Kutyniok and S. Li in connection with distributed processing and are related to the construction of global frames from local frames. In this paper we give new results on the duality of fusion frames in Hilbert spaces.     

Theory, implementation and applications of nonstationary Gabor frames

Signal analysis with classical Gabor frames leads to a fixed time-frequency resolution over the whole time-frequency plane. To overcome the limitations imposed by this rigidity, we propose an extension of Gabor theory that leads to the construction of frames with time-frequency resolution changing over time or frequency. We describe the construction of the resulting nonstationary Gabor frames and give the explicit formula for the canonical dual frame for a particular case, the painless case. We show that wavelet transforms, constant-Q transforms and more general filter banks may be modeled in the framework of nonstationary Gabor frames. Further, we present the results in the finite-dimensional case, which provides a method for implementing the above-mentioned transforms with perfect reconstruction. Finally, we elaborate on two applications of nonstationary Gabor frames in audio signal processing, namely a method for automatic adaptation to transients and an algorithm for an invertible constant-Q transform.     

Theory,implementation and applications of nonstationary Gabor frames

Signal analysis with classical Gabor frames leads to a fixed time–frequency resolution over the whole time–frequency plane. To overcome the limitations imposed by this rigidity, we propose an extension of Gabor theory that leads to the construction of frames with time–frequency resolution changing over time or frequency. We describe the construction of the resulting nonstationary Gabor frames and give the explicit formula for the canonical dual frame for a particular case, the painless case. We show that wavelet transforms, constant-Q transforms and more general filter banks may be modeled in the framework of nonstationary Gabor frames. Further, we present the results in the finite-dimensional case, which provides a method for implementing the above-mentioned transforms with perfect reconstruction. Finally, we elaborate on two applications of nonstationary Gabor frames in audio signal processing, namely a method for automatic adaptation to transients and an algorithm for an invertible constant-Q transform.     

Porosity of Mutually Nearest and Mutually Furthest Points Problems in Geodesic Space

The notion of superframe in general Hilbert spaces was introduced in the context of multiplexing, which has been widely used in mobile communication network, satellite communication network and computer area network. The notion of oblique dual frame is a generalization of conventional dual frame. It has provided us with a frame-like expansion. Using oblique dual frames one can extend frame expansions to include redundant expansions in which the analysis and synthesis frames lie in different spaces. Given positive integers L, M and N, an N?-periodic set 𝕊 in ?, let 𝒢(g, N, M) be a frame for l ²(𝕊, ? ^L ), and let 𝒢(h, N, M) be a frame for ?(h, N, M) (generated by 𝒢(h, N, M)). This article addresses super Gabor duals of g in ?(h, N, M). We obtain a necessary and sufficient condition on h admitting super oblique Gabor duals of g, and present a parametrization expression of all super oblique Gabor duals and all oblique canonical Gabor duals of g. We also characterize the uniqueness of super oblique Gabor dual and oblique canonical Gabor dual of g. Some examples are also provided.     

Generalized super Gabor duals with bounded invertible operators

In this paper, we introduce generalized super Gabor duals with bounded invertible operators by combining ideas concerning super Gabor frames with the idea of g-duals as proposed by Dehgham and Fard in 2013. Given a super Gabor frame and a bounded invertible operator A, we characterize its generalized super Gabor duals with A, and derive a parametric expression of all its generalized super Gabor duals with A. The perturbation of generalized super Gabor duals is considered as well.     

A pair of orthogonal frames

We start with a characterization of a pair of frames to be orthogonal in a shift-invariant space and then give a simple construction of a pair of orthogonal shift-invariant frames. This is applied to obtain a construction of a pair of Gabor orthogonal frames as an example. This is also developed further to obtain constructions of a pair of orthogonal wavelet frames.     

框架的局部化

Grochenig and Balan, Casazza, Heil, and Landau introduced the concepts of localization. The concepts were used to Gabor frames, wavelet frames and sampling theorem in recent years. Here they are applied to the frame of exponential windows with the conclusion that the frame of exponential windows is a Banach frame for a kind of Banach spaces, and the conclusion is also obtained about the relationship between frame bounds, frame density, measure and density of indexing set.     

不规则多生成子Gabor框架及其对偶

对给定的φ_0,…,φ_(r-1)∈L~2(R)和a_0,b_0,…,a_(r-1)1,b_(r-1)>0,本文考虑不规则多生成子Gabor系统{E_(mb_l)T_(na_l)φ_l,m,n∈Z,l=0,…,r-1}.本文给出了该系统成为L~2(R)框架的充要条件;得到了不规则多生成子Gabor框架与其对偶之间关系的刻画.特别地,给出了一类多生成子Gabor框架及其对偶的显式构造.     

Some New Inequalities for Wavelet Frames on Local Fields

Wavelet frames have gained considerable popularity during the past decade,primarily due to their substantiated applications in diverse and widespread fields of science and engineering.Finding general and verifiable conditions which imply that the wavelet systems are wavelet frames is among the core problems in time-frequency analysis.In this article,we establish some new inequalities for wavelet frames on local fields of positive characteristic by means of the Fourier transform.As an application,an improved version of the Li-Jiang inequality for wavelet frames on local fields is obtained.     

Shrinking and boundedly complete Schauder frames in Fréchet spaces

We study Schauder frames in Fréchet spaces and their duals, as well as perturbation results. We define shrinking and boundedly complete Schauder frames on a locally convex space, study the duality of these two concepts and their relation with the reflexivity of the space. We characterize when an unconditional Schauder frame is shrinking or boundedly complete in terms of properties of the space. Several examples of concrete Schauder frames in function spaces are also presented.