TWO CONDITIONS FOR NONUNIFORM GABOR FRAMES

A necessary condition is given for general nonuniform Gabor frames, which generalizes Benedetto and Walnut＇s theorem. A sufficient and necessary condition for a class of nonuniform Gabor frames is proved.

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.

A Simple Construction of Gabor Frames for L^2（R）

In this paper, we give a method which aUows us to construct a class of Parseval frames for L2（R） from Fourier frame for L2（X）. The result shows that the function which generates a Oabor frame by translations and modulations has ＂good＂ properties, i.e., it is suifficiently smooth and compactly supported.

Subspace dual super wavelet and Gabor frames

Due to its potential applications in multiplexing techniques, the study of superframes has interested some researchers. This paper addresses dual super wavelet and Gabor frames in the subspace setting. We obtain a basic-equation characterization for subspace dual super wavelet and Gabor frames. In addition, applying this characterization, we derive a procedure that allows for constructing subspace dual super wavelet frames based on certain subspace dual super Gabor frames, and vice versa. Our results are new even in L^2(R, CL) setting.

Multi-window Gabor frames and oblique Gabor duals on discrete periodic sets

Given L, N, M ∈ N and an NZ-periodic set S in Z, let l2(S) be the closed subspace of l2(Z) consisting of sequences vanishing outside S. For f = { fl : 0≤l≤L-1 }l2(Z), we denote by G(f, N, M) the Gabor system generated by f, and by L(f, N, M) the closed linear subspace generated by G(f, N, M). This paper addresses density results, frame conditions for a Gabor system G(g, N, M) in l2(S), and Gabor duals of the form G(a, N, M) in some L(h, N, M) for a frame G(g, N, M) in l2(S) (so-called oblique duals). We obtain a density theorem for a Gabor system G(g, N, M) in l2(S), and show that such condition is suficient for theexistence of g={XE1:0≤l≤L-1} with G(g,N,m) We characterize g with G(g,N,m) being respectively a frame for L(g,N,m) being a tight frame for l2(S).and G(h, N, M ) in L(h, N, M ), we establish a criterion for the existence of an oblique Gabor dual of g in L(h, N, M), study the uniqueness of oblique Gabor dual, and derive an explicit expression of a class of oblique Gabor duals (among which the one with the smallest norm is obtained). Some examples are also provided.

Necessary conditions for Gabor frames

The goal of this paper is to establish a set of necessary conditions for Gabor frames. These conditions are also sufficient for tight frames.

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).

On the Frame Set for the 3-Spline

This paper investigates the fine structure of the Gabor frame generated by the B-spline B3. In other words, one extends the known part of the Gabor frame set for the 3-spline with the construction of the compactly supported dual windows. The frame set of the function B3 is the subset of all parameters (a,b) ∈R2+ for which the time-frequency shifts of B3 along aZ × bZ form a Gabor frame for L2(R).

Density Results for Subspace Multiwindow Gabor Systems in the Rational Case

Let S be a periodic set in R and L2（S） be a subspace of L2（R）. This paper investigates the density problem for multiwindow Gabor systems in L2（S） for the case that the product of time- frequency shift parameters is a rational number. We derive the density conditions for a multiwindow Gabor system to be complete （a frame） in L2（S）. Under such conditions, we construct a multiwindow tight Gabor frame for L2 （S） with window functions being characteristic functions. We also provide a characterization of a multiwindow Gabor frame to be a Riesz basis for L2（S）, and obtain the density condition for a multiwindow Gabor Riesz basis for L2 （S）.

Expansion of Frames to Tight Frames

We show that every Bessel sequence （and therefore every frame） in a separable Hilbert space can be expanded to a tight frame by adding some elements. The proof is based on a recent generalization of the frame concept, the g-frame, which illustrates that g-frames could be useful in the study of frame theory. As an application, we prove that any Gabor frame can be expanded to a tight frame by adding one window function.

Basic Results of Gabor Frame on Local Fields

Basic facts for Gabor frame {Eu（m）bTu（n）ag}m, n∈P on local field are investigated. Accurately, that the canonical dual of frame {Eu（m）bTu（n）ag}m,n∈P also has the Gabor structure is showed; that the product ab decides whether it is possible for {Eu（m）bTu（n）ag}m,n∈P to be a frame for L^2（K） is discussed; some necessary conditions and two sufficient conditions of Gabor frame for L^2 （K） are established. An example is finally given.

Multi-window dilation-and-modulation frames on the half real line

Wavelet and Gabor systems are based on translation-and-dilation and translation-and-modulation operators,respectively,and have been studied extensively.However,dilation-and-modulation systems cannot be derived from wavelet or Gabor systems.This study aims to investigate a class of dilation-and-modulation systems in the causal signal space L^2(R+).L^2(R+)can be identified as a subspace of L^2(R),which consists of all L^2(R)-functions supported on R+but not closed under the Fourier transform.Therefore,the Fourier transform method does not work in L^2(R+).Herein,we introduce the notion ofΘa-transform in L^2(R+)and characterize the dilation-and-modulation frames and dual frames in L^2(R+)using theΘa-transform;and present an explicit expression of all duals with the same structure for a general dilation-and-modulation frame for L^2(R+).Furthermore,it has been proven that an arbitrary frame of this form is always nonredundant whenever the number of the generators is 1 and is always redundant whenever the number is greater than 1.Finally,some examples are provided to illustrate the generality of our results.

Linear independence of a finite set of time-frequency shifts

This paper introduces an open conjecture in time-frequency analysis on the linear independence of a finite set of time-frequency shifts of a given L2 function.Firstly,background and motivation for the conjecture are provided.Secondly,the main progress of this linear independence in the past twenty five years is reviewed.Finally,the partial results of the high dimensional case and other cases for the conjecture are briefly presented.

The frame sets of the B-splines

is not completely clear which elements constitute the frame sets of the B-splines currently,but some considerable results have been obtained.In this paper,firstly,the background of frame set is introduced.Secondly,the main progress of the frame sets of the B-splines in the past more than twenty years are reviewed,and particularly the progress for the frame set of the 2 order Bspline and the frame set of the 3 order B-spline are explained,respectively.