On the Galerkin Method Based on a Particular Class of Scaling Functions

The aim of this paper is to provide a large class of scaling functions for which the convergence analysis for the Galerkin method developed in [9] is applicable, whereas in that paper the only scaling functions considered for practical applications are B-splines and a few of the orthonormal Daubechies scaling functions. The functions considered here, were recently introduced in [12] where it was proved that they satisfy many properties making them interesting for the applications. In particular, here we show that the use of these functions has some advantages with respect to other basis functions.     

A Unified Approach to Construct a Class of Daubechies Orthogonal Scaling Functions

We use Lorentz polynomials to give an efficient way to prove Daubechies’ results on the existence of spline type orthogonal scaling functions and to evaluate a class of Daubechies scaling functions in a unified approach.     

Approximation properties of wavelets and relations among scaling moments II

A new orthonormality condition for scaling functions is derived. This condition shows a close connection between orthonormality and relations among discrete scaling moments. This new condition in connection with certain approximation properties of scaling functions enables to prove new relations among discrete scaling moments and consequently the same relations for continuous scaling moments.     

二维3带插值型的尺度函数

1引言众所周知,在实际应用中用小波处理信号时,小波的对称性能使信号避免失真,小波的紧支撑性使得快速小波变换的和是有限和,小波的正交性能够保持能量等等,因而构造     

A class of test functions for global optimization

We suggest weighted least squares scaling, a basic method in multidimensional scaling, as a class of test functions for global optimization. The functions are easy to code, cheap to calculate, and have important applications in data analysis. For certain data these functions have many local minima. Some characteristic features of the test functions are investigated.This paper was written while the second author was a visiting Professor at Aachen University of Technology, funded by the Deutsche Forschungsgemeinschaft.     

三进制双正交对称小波的设计

本文给出了一种三进制双正交对称小波的设计方法.在给定插值紧支撑对称尺度函数的情况下,指出了如果对偶尺度函数同为紧支撑插值的,则它们同为1-型对称.并且给出了对偶尺度函数为紧支撑插值和非插值情况下的通解计算公式.还提出了频率优化方法设计对偶尺度函数和小波函数,把双正交条件归结为线性约束的二次规划问题,最后通过线性方程组来求解.对于小波函数本文也给出了一组特解公式.     

紧支撑正交对称和反对称小波的构造

１．引言 近年来,人们分别从数学和信号的观点对正交小波进行了广泛的研究．尤其是２尺度小波,它克服了短时 Ｆｏｕｒｉｅｒ变换的一些缺陷．目前最常用的 ２尺度小波是 Ｄａｕｂｅｃｈｉｅｓ 小波,但 ２尺度小波也存在一些问题：如 Ｄａｕｂｅｃｈｉｅｓ［２］已证明了除 Ｈａａｒ小波外不存在既正交又对称的紧支撑 ２尺度小波．因此人们提出了 ａ尺度小波理论［３］－［６］,文献［４］－［６］对 ４尺度小波迸行研究．本文的目的是研究４尺度因子时紧支撑正交对称和反对称小波的构造方法．并指出对同一紧支撑正交对称尺度函数而言,…     

分数阶尺度函数的分解与重构算法

引入分数阶多分辨分析与分数阶尺度函数的概念.运用时频分析方法与分数阶小波变换,研究了分数阶正交小波的构造方法,得到分数阶正交小波存在的充要条件.给出分数阶尺度函数与小波的分解与重构算法,算法比经典的尺度函数与小波的分解与重构算法更具有一般性.     

仿射分形插值函数与纵向尺度因子

文中对一类仿射分形插值函数,用纵向尺度因子刻画了函数在特定值域分布的充分必要条件.     

三元双正交小波滤波器的刻画

研究由三元双正交插值尺度函数构造对应的双正交小波滤波器的矩阵扩充问题.当给定的一对三元双正交尺度函数中有一个为插值函数时,利用提升思想与矩阵多相分解方法,给出一类三元双正交小波滤波器的显示构造公式和一个计算实例.讨论了三元双正交小波包的的性质.     

一类完全重构滤波器的精度条件

本文研究了如何构造满足精度条件的小波滤波器.利用频域分析方法,给出了关于尺度函数精度条件的新结果以及满足精度条件的几个例子.     

对称尺度函数的参数化设计

In this paper, the factorization for filters with length Km of scaling functions intosimple blocks is considered.     

M带含参数对称正交插值尺度函数的构造

<正>1引言基于多分辨分析方法,已经构造了大量各种各样的单小波,如紧支撑的正交的Daubechies小波,紧支撑半正交的样条小波等.Daubechies证明了除Haar小波外不存在紧支撑正交对称的2带单小波.为了弥补2带单小波的不足,许多数学工作者将它进行推广得到了:双正交小波、多小波、多带小波等几大分支.而在实际应用中有时需要处理一些具有相对窄的带宽的高频信号,所以很有必要研究多带小波.同时多带小波能同时拥     

Exponentially decaying wavelets with uncertainty constants uniformly bounded with respect to the smoothness parameter

We construct a family of wavelet functions whose scaling functions decay exponentially and have uncertainty constants uniformly bounded with respect to the smoothness parameter.     

Fast and exact synthesis of some operator scaling Gaussian random fields

Operator scaling Gaussian random fields, as anisotropic generalizations of self-similar fields, know an increasing interest for theoretical studies in the literature. However, up to now, they were only defined through stochastic integrals, without explicit covariance functions. In this paper we exhibit explicit covariance functions, as anisotropic generalizations of fractional Brownian fields ones, and define corresponding Operator scaling Gaussian random fields. This allows us to propose a fast and exact method of simulation in dimension 2 based on the circulant embedding matrix method, following ideas of Stein [34] for fractional Brownian surfaces syntheses. This is a first piece of work to popularize these models in anisotropic spatial data modeling.     

Symmetric orthonormal scaling functions and wavelets with dilation factor 4

It is well known that in the univariate case, up to an integer shift and possible sign change, there is no dyadic compactly supported symmetric orthonormal scaling function except for the Haar function. In this paper we are concerned with the construction of symmetric orthonormal scaling functions with dilation factor d=4. Several examples of such orthonormal scaling functions are provided in this paper. In particular, two examples of C ¹ orthonormal scaling functions, which are symmetric about 0 and 1/6, respectively, are presented. We will then discuss how to construct symmetric wavelets from these scaling functions. We explicitly construct the corresponding orthonormal symmetric wavelets for all the examples given in this paper. This revised version was published online in June 2006 with corrections to the Cover Date.     

Daubechies Wavelets on Intervals with Application to BVPS

In this paper, Daubechies wavelets on intervals are investigated. An analytic technique for evaluating various types of integrals containing the scaling functions is proposed; they are compared with classical techniques. Finally, these results are applied to two-point boundary value problems.     

Parameter identification of 1D fractal interpolation functions using bounding volumes

Fractal interpolation functions are very useful in capturing data that exhibit an irregular (non-smooth) structure. Two new methods to identify the vertical scaling factors of such functions are presented. In particular, they minimize the area of the symmetric difference between the bounding volumes of the data points and their transformed images. Comparative results with existing methods are given that establish the proposed ones as attractive alternatives. In general, they outperform existing methods for both low and high compression ratios. Moreover, lower and upper bounds for the vertical scaling factors that are computed by the first method are presented.