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.     

Stability of Generalized Transition Fronts

We study the qualitative properties of the generalized transition fronts for the reaction–diffusion equations with the spatially inhomogeneous nonlinearity of the ignition type. We show that transition fronts are unique up to translation in time and are globally exponentially stable for the solutions of the Cauchy problem. The results hold for reaction rates that have arbitrary spatial variations provided that the rate is uniformly positive and bounded from above.     

矩阵系统的稳定与条件稳定

一个矩阵对应一种状态变换 .一系列的矩阵构成矩阵序列系统 .任一微小的扰动都会破坏伪逆的精确性 ,即状态的最后结果 .本文证明 :若矩阵系统是广义各态历经的 ,则系统是稳定的 ;若矩阵系统是普真的 ,则系统是条件稳定的 .     

The Stability of Generalized Solution of Integral Equation

When we seek the generalized solution ofthe second knid Fredholm integral equation,the integral domain is always perturbed.Is the generalized solution of the second kindFredholm integral equation stable,when the integral domain is perturbed?Before an-swering this question let us look at the following example.　 Example1　Consider the integral equationψ(x) -∫baex- yψ(y) dy =f(x) ,　　 x∈ [a,b] . (1 )Obviously,the solutionψ(x) of the equation(1 ) must satisfy thatψ(x) =f(x) + cex, (2 )wh…     

Stability of generalized monotone maps with respect to their characterizations

We show that the known types of generalized monotone maps are not stable with respect to their characterizations (i.e. the characterizations are not maintained if an arbitrary map of this type is disturbed by an element with sufficiently small norm) and introduce s-quasimonotone maps, which are stable with respect to their characterization. For gradient maps, s-quasimonotonicity is related to s-quasiconvexity (introduced by Phu in Optimization, 38, 1996) of the underlying function. A necessary and sufficient condition for a univariate polynomial to be s-quasimonotone is given. Furthermore, some stability properties of s-quasiconvex functions are presented.     

模糊Logistic方程的稳定性

研究了具有广义Hukuhara导数的模糊Logistic方程,将模糊Logistic方程转化成两个常微分方程,分析了模糊Logistic方程平衡点的稳定性.结论表明,模糊Logistic方程平衡点的稳定性不同于确定的Logistic方程,丰富了模糊微分方程研究的内容.     

指数—2非线性广义系统的解及其稳定性

本文主要给出指数－2的非线性广义系统的可解性和零解的稳定性分析．     

Stability of Gabor frames with arbitrary sampling points

Summary We study the stability of Gabor frames with arbitrary sampling points in the time-frequency plane, in several aspects. We prove that a Gabor frame generated by a window function in the Segal algebra S₀(R^d) remains a frame even if (possibly) all the sampling points undergo an arbitrary perturbation, as long as this is uniformly small. We give explicit stability bounds when the window function is nice enough, showing that the allowed perturbation depends only on the lower frame bound of the original family and some qualitative parameters of the window under consideration. For the perturbation of window functions we show that a Gabor frame generated by any window function with arbitrary sampling points remains a frame when the window function has a small perturbation in S₀(R^d) sense. We also study the stability of dual frames, which is useful in practice but has not found much attention in the literature. We give some general results on this topic and explain consequences to Gabor frames.     

广义框架的不相交性

Hilbert空间中框架的不相交性是由Han和Larson首先提出的,它与超框架有着密切的联系,在超框架及框架的构造中扮演重要的角色.广义框架是通常框架的推广.该文利用超广义框架刻画了广义框架的不相交性、强不相交性及弱不相交性;借助所得结果,给出了已知定理的不同证明方法;建立了对偶广义框架之间强不相交相和弱不相交性之间的关系;最后利用给定的强不相交的广义框架构造新的(超)对偶广义框架,所得结论恢复已有结果.     

Stability of a Jensen type equation in the space of generalized functions

We reformulate and solve the stability problem of a Jensen type functional equation

     

Dual frames in connected with generalized sampling in shift-invariant spaces

The aim of this article is to derive stable generalized sampling in a shift-invariant space by using some special dual frames in L²(0,1). These sampling formulas involve samples of filtered versions of the functions in the shift-invariant space. The involved samples are expressed as the frame coefficients of an appropriate function in L²(0,1) with respect to some particular frame in L²(0,1). Since any shift-invariant space with stable generator is the image of L²(0,1) by means of a bounded invertible operator, our generalized sampling is derived from some dual frame expansions in L²(0,1).     

Generalized sampling in shift-invariant spaces with multiple stable generators

The aim of this paper is to derive stable generalized sampling in a shift-invariant space with ? stable generators. This is done in the light of the theory of frames in the product Hilbert space (? times). The generalized samples are expressed as the frame coefficients of an appropriate function in with respect to some particular frame in . Since any multiply stable generated shift-invariant space is the image of by means of a bounded invertible operator, the generalized sampling is obtained from some dual frame expansions in . An example in the setting of the Hermite cubic splines is exhibited.     

Stability of wavelet frames with matrix dilations

Under certain assumptions we show that a wavelet frame

in remains a frame when the dilation matrices and the translation parameters are perturbed. As a special case of our result, we obtain that if is a frame for an expansive matrix and an invertible matrix , then is a frame if and for sufficiently small 0$">.

     

Nullspaces and frames

In this paper we give new characterizations of Riesz and conditional Riesz frames in terms of the properties of the nullspace of their synthesis operators. On the other hand, we also study the oblique dual frames whose coefficients in the reconstruction formula minimize different weighted norms.     

Stability of a quadratic Jensen type functional equation in the spaces of generalized functions

Making use of the fundamental solution of the heat equation we find the solution and prove the stability theorem of the quadratic Jensen type functional equation

     

g-框架的摄动和原子分解

该文完善了g-框架的稳定性理论.首先给出了g-框架摄动的几个结果,其中包括g-框架的紧摄动性.其次,讨论了g-框架和单位原子分解的关系.     

Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples

In this paper, we consider the problem of recovering a compactly supported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly. We do this by using the concept of weighted Fourier frames. A seminal result of Beurling shows that sampling points give rise to a classical Fourier frame provided they are relatively separated and of sufficient density. However, this result does not allow for arbitrary clustering of sampling points, as is often the case in practice. Whilst keeping the density condition sharp and dimension independent, our first result removes the separation condition and shows that density alone suffices. However, this result does not lead to estimates for the frame bounds. A known result of Gröchenig provides explicit estimates, but only subject to a density condition that deteriorates linearly with dimension. In our second result we improve these bounds by reducing the dimension dependence. In particular, we provide explicit frame bounds which are dimensionless for functions having compact support contained in a sphere. Next, we demonstrate how our two main results give new insight into a reconstruction algorithm—based on the existing generalized sampling framework—that allows for stable and quasi-optimal reconstruction in any particular basis from a finite collection of samples. Finally, we construct sufficiently dense sampling schemes that are often used in practice—jittered, radial and spiral sampling schemes—and provide several examples illustrating the effectiveness of our approach when tested on these schemes.     

Stability of the multi-Jensen equation

In this paper we study the stability, in the sense of Th.M. Rassias, of multi-Jensen equation.     

Exact g-frames in Hilbert spaces

G-frames, which were considered recently as generalized frames in Hilbert spaces, have many properties similar to those of frames, but not all the properties are similar. For example, exact frames are equivalent to Riesz bases, but exact g-frames are not equivalent to g-Riesz bases. In this paper, we firstly give a characterization of an exact g-frame in a complex Hilbert space. We also obtain an equivalent relation between an exact g-frame and a g-Riesz basis under some conditions. Lastly we consider the stability of an exact g-frame for a Hilbert space under perturbation. These properties of exact g-frames for Hilbert spaces are not similar to those of exact frames.     

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.