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.     

The Theory of Multiresolution Analysis Frames and Applications to Filter Banks

The notion of a frame multiresolution analysis (FMRA) is formulated. An FMRA is a natural extension to affine frames of the classical notion of a multiresolution analysis (MRA). The associated theory of FMRAs is more complex than that of MRAs. A basic result of the theory is a characterization of frames of integer translates of a function φ in terms of the discontinuities and zero sets of a computable periodization of the Fourier transform of φ. There are subband coding filter banks associated with each FMRA. Mathematically, these filter banks can be used to construct new frames for finite energy signals. As with MRAs, the FMRA filter banks provide perfect reconstruction of all finite energy signals in any one of the successive approximation subspacesV_jdefining the FMRA. In contrast with MRAs, the perfect reconstruction filter bank associated with an FMRA can be narrow band. Because of this feature, in signal processing FMRA filter banks achieve quantization noise reduction simultaneously with reconstruction of a given narrow-band signal.     

连续信号和离散信号小波多分辨率分析的实现

本文首先以信号分析为背景阐述多分辨率分析的基本思想,然后从多分辨率分析的角度,研究连续信号小波变换的特点及实现方法,并对离散信号的多分辨率分解和重构进行讨论。文中还就相应的一些概念和问题展开了讨论,并提出了一些看法。     

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.     

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.     

Adelic Multiresolution Analysis, Construction of Wavelet Bases and Pseudo-Differential Operators

In our previous paper, the Haar multiresolution analysis (MRA) $\{V_{j}\}_{j\in \mathbb {Z}}$ in $L^{2}(\mathbb {A})$ was constructed, where $\mathbb {A}$ is the adele ring. Since $L^{2}(\mathbb {A})$ is the infinite tensor product of the spaces $L^{2}({\mathbb {Q}}_{p})$ , p=∞,2,3,…, the adelic MRA has some specific properties different from the corresponding finite-dimensional ones. Nevertheless, this infinite-dimensional MRA inherits almost all basic properties of the finite-dimensional case. In this paper we derive explicit formulas for bases in V _j , $j\in \mathbb {Z}$ , and for the wavelet bases generated by the above-mentioned adelic MRA. In view of the specific properties of the adelic MRA, there arise some technical problems in the construction of wavelet bases. These problems were solved with the aid of the operator formalization of the process of generation of wavelet bases. We study the spectral properties of the fractional operator introduced by S. Torba and W.A. Zúñiga-Galindo. We prove that the constructed wavelet functions are eigenfunctions of this fractional operator. This paper, as well as our previous paper, introduces new ideas to construct different infinite-dimensional MRAs. Our results can be used in the theory of adelic pseudo-differential operators and equations over the ring of adeles and in adelic models in physics.     

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 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( ).     

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 Frames and the Density and Connectivity of Wavelet Frames

We examine some recent results of Bownik on density and connectivity of the wavelet frames. We use orthogonality (strong disjointness) properties of frame and Bessel sequences, and also properties of Bessel multipliers (operators that map wavelet Bessel functions to wavelet Bessel functions). In addition we obtain an asymptotically tight approximation result for wavelet frames.     

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   

p-Adic Pseudodifferential Operators and p-Adic Wavelets

We introduce a new wide class of p-adic pseudodifferential operators. We show that the basis of p-adic wavelets is the basis of eigenvectors for the introduced operators.     

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.     

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.     

Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

This paper generalizes the mixed extension principle in L ₂(ℝ ^d ) of (Ron and Shen in J. Fourier Anal. Appl. 3:617–637, 1997) to a pair of dual Sobolev spaces H ^s (ℝ ^d ) and H ^−s (ℝ ^d ). In terms of masks for φ,ψ ¹,…,ψ ^L ∈H ^s (ℝ ^d ) and , simple sufficient conditions are given to ensure that (X ^s (φ;ψ ¹,…,ψ ^L ), forms a pair of dual wavelet frames in (H ^s (ℝ ^d ),H ^−s (ℝ ^d )), where

Orthogonal and Nonorthogonal Multiresolution Analysis, Scale Discrete and Exact Fully Discrete Wavelet Transform on the Sphere

Based on a new definition of dilation a scale discrete version of spherical multiresolution is described, starting from a scale discrete wavelet transform on the sphere. Depending on the type of application, different families of wavelets are chosen. In particular, spherical Shannon wavelets are constructed that form an orthogonal multiresolution analysis. Finally fully discrete wavelet approximation is discussed in the case of band-limited wavelets. June 18, 1996. Date revised: January 14, 1997.     

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.     

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.