基于小波神经网络的NLAR(p)过程的逼近

小波神经网络是近年来发展起来的一种逼近非线性函数的新型人工神经网络。特别是，正交尺度函数为某函数的小波神经网络更适合于函数逼近。本文在此基础上讨论了小波神经网络对非线性AR(p)过程的逼近。     

基于高斯型窗函数的基小波构造

阐述了基于高斯型窗函数的可容基小波构造,讨论了若干类基小波.首先引入若干经典基小波如墨西哥草帽小波、莫莱小波、DOG犬小波和盖博解析小波,作者发现它们具有统一的结构,即均由高斯窗函数生成;进而在犬小波结构的启示下,构造了由高斯窗函数的差形成的犬小波族,对之验证了可容性条件;并且将它推广为有限个高斯窗函数的线性组合形成的小波,确定了带通条件.     

Homogeneous approximation property for wavelet frames with matrix dilations

The homogeneous approximation property (HAP) for wavelet frames was studied recently. The HAP is useful in practice since it means that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time‐scale shifts. In this paper, we prove the HAP for wavelet frames generated by admissible wavelet functions with arbitrary translation parameters and a class of dilation matrices. Moreover, we show that the approximation is uniform to some extent whenever wavelet functions satisfy moderate smooth and decaying conditions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evaluation of a continuous wavelet transform by solving the Cauchy problem for a system of partial differential equations

It is shown that the problem of evaluating the continuous Morlet wavelet transform can be stated as the Cauchy problem for a system of two partial differential equations. The initial conditions for the desired functions, i.e., for the real and imaginary parts of the wavelet transform, are the analyzed function and a vanishing function, respectively. Numerical examples are given.     

Tree wavelet approximations with applications

We construct a tree wavelet approximation by using a constructive greedy scheme (CGS). We define a function class which contains the functions whose piecewise polynomial approximations generated by the CGS have a prescribed global convergence rate and establish embedding properties of this class. We provide sufficient conditions on a tree index set and on bi-orthogonal wavelet bases which ensure optimal order of convergence for the wavelet approximations encoded on the tree index set using the bi-orthogonal wavelet bases. We then show that if we use the tree index set associated with the partition generated by the CGS to encode a wavelet approximation, it gives optimal order of convergence.     

Convergence of Riemannian sums of inverse wavelet transforms

We study the approximation of the inverse wavelet transform using Riemannian sums.We show that when the Fourier transforms of wavelet functions satisfy some moderate decay condition,the Riemannian sums converge to the function to be reconstructed as the sampling density tends to infinity.We also study the convergence of the operators introduced by the Riemannian sums.Our result improves some known ones.     

Correlation Wavelet and its Applications

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有限区间内四阶样条小波的构造

用有限区间上的截断4阶B样条,构造了有限区间上的4阶样条小波。这些小波由边界小波和内部小波组成,对某一尺度,它们组成了有限维的小波空间。于是,任何有限区间上的函数皆可表示为该区间上的尺度函数和小波函数的有限和,即小波级数,这克服了用无穷区间上的小波进行有限信号处理时,在边界上误差较大的不足,同时将该小波用于偏微分方程具有同样重要的意义。     

Wavelets and framelets from dual pseudo splines

Dual pseudo splines constitute a new class of refinable functions with B-splines as special examples.In this paper,we shall construct Riesz wavelet associated with dual pseudo splines.Furthermore,we use dual pseudo splines to construct tight frame systems with desired approximation order by applying the unitary extension principle.     

The construction of plane elastomechanics and Mindlin plate elements of B-spline wavelet on the interval

Based on two-dimensional tensor product B-spline wavelet on the interval (BSWI), a class of C0 type plate elements is constructed to solve plane elastomechanics and moderately thick plate problems. Instead of traditional polynomial interpolation, the scaling functions of two-dimensional tensor product BSWI are employed to form the shape functions and construct BSWI elements. Unlike the process of direct wavelets adding in the previous work, the elemental displacement field represented by the coefficients of wavelets expansions is transformed into edges and internal modes via the constructed transformation matrix in this paper. The method combines the versatility of the conventional finite element method (FEM) with the accuracy of B-spline functions approximation and various basis functions for structural analysis. Some numerical examples are studied to demonstrate the proposed method and the numerical results presented are in good agreement with the closed-form or traditional FEM solutions.     

New construction of wavelets base on floor function

In this paper, the properties of the floor function has been used to find a function which is one on the interval [0, 1) and is zero elsewhere. The suitable dilation and translation parameters lead us to get similar function corresponding to the interval [a,b). These functions and their combinations enable us to represent the stepwise functions as a function of floor function. We have applied this method on Haar wavelet, Sine–Cosine wavelet, Block-Pulse functions and Hybrid Fourier Block-Pulse functions to get the new representations of these functions.     

Pointwise convergence of a class of non-orthogonal wavelet expansions

Non-orthogonal wavelet expansions associated with a class of mother wavelets is considered. This class of wavelets comprises mother wavelets that are not necessarily integrable over the whole real line, such as Shannon's wavelet. The pointwise convergence of these wavelet expansions is investigated. It is shown that, unlike other wavelet expansions, the ones under consideration do not necessarily converge pointwise to the functions at points of continuity, unless a more stringent condition, such as bounded variation, is imposed.

     

Construction of periodic wavelet frames with dilation matrix

An important tool for the construction of periodic wavelet frame with the help of extension principles was presented in the Fourier domain by Zhang and Saito [Appl. Comput. Harmon. Anal., 2008, 125： 68-186]. We extend their results to the dilation matrix cases in two aspects. We first show that the periodization of any wavelet frame constructed by the unitary extension principle formulated by Ron and Shen is still a periodic wavelet frame under weaker conditions than that given by Zhang and Saito, and then prove that the periodization of those generated by the mixed extension principle is also a periodic wavelet frame if the scaling functions have compact supports.     

Hamilton-Jacobi方程的小波Galerkin方法

本文选择Daubechies小波尺度函数空间作为Galerkin方法的测试函数空间,并将其应用于Hamilton-Jacobi方程,得到了求解Hamilton-Jacobi方程的小波Galerkin方法的数值格式．由于小波在时间和频率上的局部性,本方法适用于处理具有奇异解的问题,可以有效地防止数值振荡．数值试验显示,本方法是有效的．     

Analytic wavelets generated by radial functions

In this paper, we deal with a class of non-stationary multiresolution analysis and wavelets generated by certain radial basis functions. These radial basis functions are noted for their effectiveness in terms of projection, such as interpolation and least-squares approximation, particularly when the data structure is scattered or the dimension of ^s is large. Thus projecting a functionf onto a suitable multiresolution space is relatively easy here. The associated multiresolution spaces approximate sufficiently smooth functions exponentially fast. The non-stationary wavelets satisfy the Littlewood-Paley identity so that perfect reconstruction of wavelet decompositions is achieved. For the univariate case, we give a detailed analysis of the time-frequency localization of these wavelets. Two numerical examples for the detection of singularities with analytic wavelets are provided.Supported by NSF Grant #DMS-92-06928, Air Force Grant #F49620-92-J-0403DEF and NATO Grant #CR-900158.     

再生核与小波变换

在再生核基本理论的基础上,介绍了再生核在小波变换中的作用,并且根据连续小波变换像空间是再生核Hilbert空间这一基本事实,借助再生核理论的特殊技巧,建立了Littlewood-Paley和Haar小波变换像空间的再生核函数与已知再生核空间的再生核的关系,为小波变换像空间的进一步研究提供理论基础.     

人类胚胎期小脑组织基因表达的小波多尺度分析

对人类胚胎期小脑组织基因基于时空的表达强度进行小波多尺度5层分解与重构,在常用小波函数中,db2的去噪效果最好,可以用于小脑组织基因表达强度的去噪处理.     

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.