局部域上Gabor紧框架的特征

主要讨论局部域上的Gabor紧框架.首先,建立局部域上Gabor系{xm(bx)g(x-u(n)a)}m.n∈p构成L~2(K)上紧框架的特征.其次,给出Gabor系{X_m(bx)g(x-u(n)a)}_(m,n∈p)成为L~2(K)上标准正交基的充要条件.     

非参数回归函数最近邻估计LP收敛速度

本文拟给出非参数回归函数最近邻估计L_P收敛速度的一般性结果,同时把韦来生等的结果(见文[1])作为本文结果的一种特殊情形。本文的证明思路源于文[1]。 我们仔细研究了文[1]的证明过程,发现文[1]的主要定理(后称定理)的条件“m(x)适合阶为1的lipschitz条件”可减弱为“m(x)在(A为指标集,θ_j为R~d中的     

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.     

L2(Rd)的Gabor框架的扰动

本文研究了L2(Rd)上以矩阵平移和调制的Gabor框架的扰动,得到了若干有意义的结果.     

实直线周期子集上的向量值子空间弱Gabor双框架

因其在多路复用技术中的潜在应用,超框架(又称向量值框架)和子空间框架受到了众多数学家和工程专家的关注.弱双框架是希尔伯特空间中双框架的推广.本文研究实直线周期子集上的向量值子空间弱Gabor双框架(WGBFs),即L~2(S,C~L)中的WGBFs,其中S是R上的周期子集.利用Zak变换矩阵方法,得到了WGBFs的刻画,它将构造WGBFs的问题归结为设计有限阶Zak变换矩阵;给出了WGBFs的一个例子定理;导出了WGBFs的一个稠密性定理.     

空间n边形中的射影定理和余弦定理

三角形中的射影定理、余弦定理和正弦定理,文[1]已(于1954年)推证到凸n边形。文[2]则应用不同的方法(复数方法)对文[1]的结论进行了再论证。文[3]将前两个定理推证到n面体。本文拟应用向量代数中的一个最基本的等式推证,较易得到空间n边形中的射影定理和余弦定理。     

Banach空间的框架和原子分解的摄动及其稳定性

本文利用框架理论对 Banach空间的框架和原子分解的摄动及其稳定性进行了研究,改进并推广了文[5,6]的工作.     

扰动双中心二次系统的全局分枝与浑沌性

本文考虑二次系统(dx)/(dt)=-cy~2+1/(4c),(dy)/(dt)=ax~2-1/(4a)(1.1)的多参数扰动.自治扰动系统与“弱化的 Hilbert 第16问题”有关,并涉及文[2]否定二次系统(2,2)极限环分布的讨论.非自治扰动系统与 P.Holmes 研究过的Duffing 振子不同,又有新的分枝性质,并且是文[4]工作的继续.顺便指出,本文所用文[3]讨论自治扰动的方法比 J.carr 等文[5]更一般化.文[5]若引用[3]中的引理,可简化其证明.     

自治系统的不稳定定理及其应用

本文建立了一个关于自治系统(2.1)的未被扰动运动为不稳定的定理,它是Красовский在文[2]中建立的不稳定定理的推广。运用这个定理,本文讨论了两个三阶非线性系统未被扰动运动为不稳定的条件,对文[3]中给出的零解不稳定条件进行了改进。     

紧支撑 Weyl-Heisenberg 框架的摄动

本文研究了L~2(R)上具有紧支撑的Weyl-Heisenberg框架分别对窗口函数、平移指标、旋转指标以及多项混合摄动的稳定性.     

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.     

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.     

非均匀Gabor框架的稳定性

研究了当窗函数变化时非均匀Gabor框架的稳定性.对紧支撑Gabor框架,将均匀情况下关于稳定性的结论推广到了非均匀的情况;对一般的Gabor框架,利用W（L^∞,e^1）范数给出了其稳定的一个充分条件.     

Deformation of Gabor systems

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 theorem requires a new characterization of Gabor frames and Gabor Riesz sequences. It is in the style of Beurling's characterization of sets of sampling for bandlimited functions and extends significantly the known characterization of Gabor frames “without inequalities” from lattices to non-uniform sets.     

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.     

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.     

Irregular Gabor frames and their stability

In this paper we give sufficient conditions for irregular Gabor systems to be frames. We show that for a large class of window functions, every relatively uniformly discrete sequence in with sufficiently high density will generate a Gabor frame. Explicit frame bounds are given. We also study the stability of irregular Gabor frames and show that every Gabor frame with arbitrary time-frequency parameters is stable if the window function is nice enough. Explicit stability bounds are given.

     

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).     

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.     

Varying the time-frequency lattice of Gabor frames

A Gabor or Weyl-Heisenberg frame for is generated by time-frequency shifts of a square-integrable function, the Gabor atom, along a time-frequency lattice. The dual frame is again a Gabor frame, generated by the dual atom. In general, Gabor frames are not stable under a perturbation of the lattice constants; that is, even for arbitrarily small changes of the parameters the frame property can be lost.

In contrast, as a main result we show that this kind of stability does hold for Gabor frames generated by a Gabor atom from the modulation space , which is a dense subspace of . Moreover, in this case the dual atom depends continuously on the lattice constants. In fact, we prove these results for more general weighted modulation spaces. As a consequence, we obtain for Gabor atoms from the Schwartz class that the continuous dependence of the dual atom holds even in the Schwartz topology. Also, we complement these main results by corresponding statements for Gabor Riesz sequences and their biorthogonal system.