Sobolev空间H~s(R~d)上波包系的框架性质

波包系是通过对有限个函数做伸缩、平移和调制三种运算生成的一种新型函数系,因此传统的小波系和Gabor系都是它的特殊情况.该文首先给出了Sobolev空间H~s(R~d)中一个广义平移函数系成为Bessel点列或框架的充分条件,然后结合波包系是一类特殊的广义平移函数系这一结果,给出了高维Sobolev空间H~s(R~d)上波包系成为框架的一个充分条件.最后,利用矩阵的特征值理论,该文证明了:如果函数g的Fourier变换在某一开球中大于某个正数,那么由它生成的波包系不能成为H~s(R~d)的一个框架.     

M-带插值小波包

本文给出M-带插值小波包的构造.M-带插值小波包是根据基插值函数建立的迭代函数序列进行伸缩平移的空间序列.这种小波包可使信号分解更为精细,并具有更好的局部性.由此建立了这种小波包子空间上的近似采样定理.     

小波紧框架的构造

小波框架理论是小波分析的重要内容之一.本文对于4-带尺度函数,由V1中的l个函数ψ1,ψ2,…,ψl构造小波紧框架.首先给出这个l个函数构成小波紧框架的充分条件.由此给出由4-带尺度函数构造出一个小波紧框架的公式.最后还给出类似于小波的小波紧框架的分解与重构算法.     

二元3带小波紧框架的构造

研究二元3带小波紧框架的结构.首先给出二元3带小波紧框架的充分条件.并给出这种小波紧框架的显式公式.若给定的尺度函数的符号函数是有理函数,则可以构造出符号函数为有理函数的小波紧框架.文中给出了数值例子,还给出了二元3带小波紧框架的分解和重构算法.     

小波紧框架的显式构造

该文研究对应于3带尺度函数的小波紧框架,这个小波紧框架是由V_1中的l个函数ψ^1, ψ^2, ψ^n 构成.给出这l个函数构成小波紧框架的充分条件.由此给出由3 带尺度函数构造出一个小波紧框架的显式公式.特别的,如果给定尺度函数的符号是有理函数,则可以构造出符号为有理函数的小波紧框架.最后还给出类似于小波的小波紧框架的分解与重构算法.      

α带小波紧框架的显式构造方法

文中研究了对应于α-带尺度函数的小波紧框架,这个小波紧框架是由V₁中的n个函数ψ¹,ψ²,...,ψⁿ构成. 首先给出了这n个函数构成小波紧框架的充分条件, 并借助尺度函数给出了构造小波紧框架的显式公式. 如果尺度函数的符号是有理函数,则可以构造出符号为有理函数的小波紧框架. 其次给出类似于正交小波的小波紧框架的分解与重构算法,并构造了小波紧框架的数值算例.     

具有整数值伸缩矩阵的三维紧框架小波的存在性

引进了三维紧框架小波的概念,它是由框架多分辨分析中子空间X_1中的若干个三维函数Γ~1(y),Γ~2(y),…,Γ~n(y)构成的.研究了对应于三维尺度函数的三维紧框架小波的存在性.运用时频分析方法、滤波器理论、算子理论,给出这n个三维函数生成小波紧框架的充分条件,得到了由一个尺度函数Ψ(y)构造三维紧框架小波的显式公式.     

N维2带小波紧框架的构造

给出了$m$个函数生成$N$维2带小波紧框架的充分条件和$N$维2带小波紧框架的显式构造算法, 讨论了小波紧框架的分解算法与重构算法. 提出的构造方法很有普遍性, 容易推广到$N(N\geq2)$维$M(M\geq 2)$带小波紧框架的情形,也可以得到类似的小波紧框架的分解算法与重构算法.     

平方可积函数空间的仿射框架

推广了双正交小波的概念.引进了多尺度平移伪框架的概念.给出了它的塔式分解格式及其存在的条件.进而得到平方可积函数空间的函数仿射伪框架展式.     

广义小波框架

通过将分数阶Fourier变换(Fr FT)与传统小波框架(WF)相结合,本文引入了一类新的框架,即广义小波框架(GWF).首先介绍了分数阶Fourier变换、级数以及相关结论;然后通过分析母小波的Fourier变换,得到了L2(R)中广义小波框架的若干个充分条件和必要条件;最后在参数选择为a=2,b=1的特殊情形下,给出了L2(R)中紧广义小波框架的一个充要条件.     

Perturbations of frames

In this paper, we give some sufficient conditions under which perturbations preserve Hilbert frames and near-Riesz bases. Similar results are also extended to frame sequences, Riesz sequences and Schauder frames. It is worth mentioning that some of our perturbation conditions are quite different from those used in the previous literatures on this topic.     

Sufficient Conditions for Irregular Wavelet Frames

Finding verifiable conditions for wavelet systems to be wavelet frames is among the core problems in wavelet analysis. In this paper, we give some simple and sufficient conditions that ensure a multidimensional irregular wavelet system to be a frame or a weighted frame. Quantitative results are provided, and explicit frame bounds are given.     

Hilbert空间中的g-Bessel序列的一些等式与不等式(II)

Hilbert 空间中的g- 框架是框架的自然推广, 它们包含了许多推广的框架, 如子空间框架或fusion 框架、斜框架和拟框架等. 它们有许多与框架类似的性质, 但是并不是所有的性质都是相似的.例如, 无冗框架等价于Riesz 基, 但是无冗g- 框架不等价于g-Riesz 基. 一些作者将Hilbert 空间中的框架和对偶框架的等式和不等式推广到g- 框架和对偶g- 框架. 本文建立Hilbert 空间中的g-Bessel序列或g- 框架的一些新的等式和不等式. 本文还给出这些不等式的等号成立的充要条件. 这些结果推广和改进了由Balan, Casazza 和G?vruta 等得到的著名结果.     

本刊英文版Vol.30(2014),No.7论文摘要(英文)

<正>Perturbations of Frames Dong Yang CHEN;;Lei LI;;Ben Tuo ZHENG Abstract In this paper,we give some sufficient conditions under which perturbations preserve Hilbert frames and near-Riesz bases.Similar results are also extended to frame sequences,Riesz sequences and Schauder frames.It is worth mentioning that some of our perturbation conditions are quite different from those used in the previous literatures on this topic.Perturbations of Moore-Penrose Metric Generalized Inverses of Linear Operators in Banach Spaces Hai Feng MA;;Shuang SUN;;Yu Wen WANG;;Wen Jing ZHENG     

Dual generators for weighted irregular wavelet frames and reconstruction error

Dual frames are very useful tools to reconstruct a function and have been explored in many different aspects. In this paper, we give the conditions under which a class of special weighted irregular wavelet frames could have a dual generator with a very explicit form. We also prove that when the irregular translations are changed in some admissible range, the reconstruction formula has a very small perturbation. An example is given to show the use of our main results.     

Sufficient conditions for irregular Gabor frames

Finding general and verifiable conditions which imply that Gabor systems are (resp. cannot be) Gabor frames is among the core problems in Gabor analysis. In their paper on atomic decompositions for coorbit spaces [H.G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations, and their atomic decomposition, I, J. Funct. Anal. 86 (1989), 307–340], the authors proved that every Gabor system generated with a relatively uniformly discrete and sufficiently dense time-frequency sequence will allow series expansions for a large class of Banach spaces if the window function is nice enough. In particular, such a Gabor system is a frame for the Hilbert space of square integrable functions. However, their proof is based on abstract analysis and does not give direct information on how to determine the density in the sense of directly applicable estimates. It is the goal of this paper to present a constructive version of the proof and to provide quantitative results. Specifically, we give a criterion for the general case and explicit density for some cases. We also study the existence of Gabor frames and show that there is some smooth window function such that the corresponding Gabor system is incomplete for arbitrary time-frequency lattices.     

g-Besselian frames in Hilbert spaces

In this paper, we introduce the concept of a g-Besselian frame in a Hilbert space and discuss the relations between a g-Besselian frame and a Besselian frame. We also give some characterizations of g-Besselian frames. In the end of this paper, we discuss the stability of g-Besselian frames. Our results show that the relations and the characterizations between a g-Besselian frame and a Besselian frame are different from the corresponding results of g-frames and frames.     

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.     

SOME PROPERTIES OF OPERATOR-VALUED FRAMES

Operator-valued frames(or g-frames) are generalizations of frames and fusion frames and have been used in packets encoding, quantum computing, theory of coherent states and more. In this article, we give a new formula for operator-valued frames for finite dimensional Hilbert spaces. As an application, we derive in a simple manner a recent result of A. Najati concerning the approximation of g-frames by Parseval ones. We obtain also some results concerning the best approximation of operator-valued frames by its alternate duals,with optimal estimates.     

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.