On Dual Wavelet Tight Frames

A characterization of multivariate dual wavelet tight frames for any general dilation matrix is presented in this paper. As an application, Lawton's result on wavelet tight frames inL²( ) is generalized to then-dimensional case. Two ways of constructing certain dual wavelet tight frames inL²( ⁿ) are suggested. Finally, examples of smooth wavelet tight frames inL²( ) andH²( ) are provided. In particular, an example is given to demonstrate that there is a function ψ whose Fourier transform is positive, compactly supported, and infinitely differentiable which generates a non-MRA wavelet tight frame inH²( ).     

Irregular Wavelet Frames and Gabor Frames

Given g∈L²(R ⁿ ), we consider irregular wavelet for the form\(\left\{ {\lambda ^{\frac{n}{2}} g\left( {\lambda _j x - kb} \right)} \right\}_{j\varepsilon zj\varepsilon z^n } ,where\;\lambda _j \) > 0 and b > 0. Sufficient conditions for the wavelet system to constitute a frame for L²(R ⁿ ) are given. For a class of functions g∈L²²(R ⁿ ) we prove that certain growth conditions on {λ _j } will frames, and that some other types of sequences exclude the frame property. We also give a sufficient condition for a Gabor system\(\left\{ {e^{zrib\left( {j,x} \right)} g\left( {x - \lambda _k } \right)} \right\}_{j\varepsilon z^n ,k\varepsilon z} \)to be a frame.     

Irregular Wavelet Frames and Gabor Frames

Given g { l^\fracn2 g( l_j x - kb ) }_jezjezⁿ ,where  l_j \left\{ {\lambda ^{\frac{n}{2}} g\left( {\lambda _j x - kb} \right)} \right\}_{j\varepsilon zj\varepsilon z^n } ,where\;\lambda _j > 0 and b > 0. Sufficient conditions for the wavelet system to constitute a frame for L ²(R ⁿ ) are given. For a class of functions g{ e^{zrib( j,x )} g( x - l_k ) }_{jezⁿ ,kez}\left\{ {e^{zrib\left( {j,x} \right)} g\left( {x - \lambda _k } \right)} \right\}_{j\varepsilon z^n ,k\varepsilon z} to be a frame.     

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.     

Super-Wavelets and Decomposable Wavelet Frames

A wavelet frame is called decomposable whenever it is equivalent to a superwavelet frame of length greater than one. Decomposable wavelet frames are closely related to some problems on super-wavelets. In this article we first obtain some necessary or sufficient conditions for decomposable Parseval wavelet frames. As an application of these conditions, we prove that for each n > 1 there exists a Parseval wavelet frame which is m-decomposable for any 1 < m ≤ n, but not k-decomposable for any k > n. Moreover, there exists a super-wavelet whose components are non-decomposable. Similarly we also prove that for each n > 1, there exists a Parseval wavelet frame that can be extended to a super-wavelet of length m for any 1 < m ≤ n, but can not be extended to any super-wavelet of length k with k > n. The connection between decomposable Parseval wavelet frames and super-wavelets is investigated, and some necessary or sufficient conditions for extendable Parseval wavelet frames are given.     

On Generating Tight Gabor Frames at Critical Density

We consider the construction of tight Gabor frames (h, a=1, b=1) from Gabor systems (g, a=1, b=1) with g a window having few zeros in the Zak transform domain via the operation h=Z ^-1(Zg/|Zg|), where Z is the standard Zak transform. We consider this operation with g the Gaussian, the hyperbolic secant, and for g belonging to a class of positive, even, unimodal, rapidly decaying windows of which the two-sided exponential is a typical example. All these windows g have the property that Zg has a single zero, viz. at (1/2,\1/2), in the unit square [0,1)². The Gaussian and hyperbolic secant yield a frame for any a,b > 0 with ab < 1, and we show that so does the two-sided exponential. For these three windows it holds that S _a ^-1/2 g h as a 1, where S _a is the frame operator corresponding to the Gabor frame (g,a,a). It turns out that the hs corresponding to gs of the above type look and behave quite similarly when scaling parameters are set appropriately. We give a particular detailed analysis of the h corresponding to the two-sided exponential. We give several representations of this h, and we show that , and is continuous and differentiable everywhere except at the half-integers, etc., and we pay particular attention to the cases that the time constant of the two-sided exponential g tends to . We also consider the cases that the time constants of the Gaussian and of the hyperbolic secant tend to 0 or to . It so turns out that h thus obtained changes from the box function into its Fourier transform when the time constant changes from 0 to      

Density of Gabor Frames

A Gabor system is a set of time-frequency shifts S(g, Λ) ={e^{2 π ibx}g(x − a)}_{(a, b) Λ} of a function g L²(R^d). We prove that if a finite union of Gabor systems _{k = 1}^rS(g_k, Λ_k) forms a frame for L²(R^d) then the lower and upper Beurling densities of Λ = _{k = 1}^r Λ_k satisfy D⁻(Λ) ≥ 1 and D ⁺ (Λ) < ∞. This extends recent work of Ramanathan and Steger. Additionally, we prove the conjecture that no collection _{k = 1}^r{g_k(x − a)}_{a Γk} of pure translates can form a frame for L²(R^d).     

Inequalities for Wavelet Frames

One of the fundamental problems in the study of wavelet frames is to find conditions on the wavelet function and the dilation and translation parameters so that the corresponding wavelet system forms a frame. In this article, we obtain some inequalities for a discrete wavelet system to be a frame. Our result improves known ones by Chui, Shi, and Chen.     

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.     

On Characterization of Nonuniform Tight Wavelet Frames on Local Fields

In this article, we introduce a notion of nonuniform wavelet frames on local fields of positive characteristic. Furthermore, we gave a complete characterization of tight nonuniform wavelet frames on local fields of positive characteristic via Fourier transform. Our results also hold for the Cantor dyadic group and the Vilenkin groups as they are local fields of positive characteristic.     

小波连通性研究进展

本文介绍了 L2(R)中一些特殊小波组成集在L2(R)中的拓扑几何性质——道路连通性,特别综述了近二十年来小波连通问题研究的主要进展并列出了一些悬而未决的问题,并简要说明了高维情形和其他情形的相应结果.     

p-Adic Multiresolution Analysis and Wavelet Frames

We study p-adic multiresolution analyses (MRAs). A complete characterization of test functions generating an MRA (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions and that all such scaling functions generate the Haar MRA. We also suggest a method for constructing sets of wavelet functions and prove that any set of wavelet functions generates a p-adic wavelet frame.     

二元多重小波紧框架的矩阵频域乘子

矩阵频域乘子是由本质有界可测函数组成的矩阵,它能将多重小波紧框架映射成多重小波紧框架.引入二元多重小波紧框架的矩阵频域乘子的概念,给出了一个矩阵值函数成为二元多重小波紧框架的矩阵乘子的充要条件,并给出了构造例子.     

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.     

On the Orthogonality of Geometric Codes

We solve some problems concerning the orthogonality of geometric codes associated with sets of i- and j-dimensional subspaces of PG(n, q). Various applications are found, and we discuss all the interesting cases in small dimensional spaces.     

不规则小波框架的膨胀列

研究了L2(R)中小波框架{ψj,k}j,k={sjψ(sj·-kb)}j,k∈Z的膨胀列{sj}j的性质.如果{ψj,k}j,k是L2(R)的一个小波框架,那么膨胀列是无界的,在某些条件下{sj}j∈Z一定能够被重排为指标集Z上的一个非减数列,而且存在常数λ,μ∈(0,1)和p∈Z ,使得对j∈Z有λ相似文献   

A Class of Bidimensional FMRA Wavelet Frames

This paper addresses the construction of wavelet frame from a frame multiresolution analysis （FMRA） associated with a dilation matrix of determinant ±2. The dilation matrices of determinant ±2 can be classified as six classes according to integral similarity. In this paper, for four classes of them, the construction of wavelet frame from an FMRA is obtained, and, as examples, Shannon type wavelet frames are constructed, which have an independent value for their optimality in some sense.     

History and Evolution of the Density Theorem for Gabor Frames

The Density Theorem for Gabor Frames is one of the fundamental results of time-frequency analysis. This expository survey attempts to reconstruct the long and very involved history of this theorem and to present its context and evolution, from the one-dimensional rectangular lattice setting, to arbitrary lattices in higher dimensions, to irregular Gabor frames, and most recently beyond the setting of Gabor frames to abstract localized frames. Related fundamental principles in Gabor analysis are also surveyed, including the Wexler-Raz biorthogonality relations, the Duality Principle, the Balian-Low Theorem, the Walnut and Janssen representations, and the Homogeneous Approximation Property. An extended bibliography is included.     

Smooth Wavelet Tight Frames with Zero Moments

This paper considers the design of wavelet tight frames based on iterated oversampled filter banks. The greater design freedom available makes possible the construction of wavelets with a high degree of smoothness, in comparison with orthonormal wavelet bases. In particular, this paper takes up the design of systems that are analogous to Daubechies orthonormal wavelets—that is, the design of minimal length wavelet filters satisfying certain polynomial properties, but now in the oversampled case. Gröbner bases are used to obtain the solutions to the nonlinear design equations. Following the dual-tree DWT of Kingsbury, one goal is to achieve near shift invariance while keeping the redundancy factor bounded by 2, instead of allowing it to grow as it does for the undecimated DWT (which is exactly shift invariant). Like the dual tree, the overcomplete DWT described in this paper is less shift-sensitive than an orthonormal wavelet basis. Like the examples of Chui and He, and Ron and Shen, the wavelets are much smoother than what is possible in the orthonormal case.     

由多重贝塞尔序列扩充为仿射框架

仿射框架在信号处理中有实用性.运用算子理论与时频分析,将两个二重贝塞尔序列扩充为一对对偶二重仿射框架.再由已知的一对多重贝塞尔序列添加若干个函数使它们扩充为一对对偶多重仿射框架,得到了多重Gabor框架的特征不等式.