ON THE CONVERGENCE OF APPROXIMATION TO CAUCHY SINGULAR INTEGRALS BY WAVELETS

本文讨论了用周期小波对Ｈｏ．．ｌｄｅｒ空间进行的特征刻划，给出了Ｈｏ．．ｌｄｅｒ空间中的若干小波逼近性质，证明了用周期小波逼近Ｃａｕｃｈｙ型奇异积分时的一些收敛定理．     

区间上的双正交小波的一种构造方法

小波分析是近十几年来十分热门的课题,早期的小波都是定义在无穷区间上的．而实际问题常常是有限区间,常用的方法是将有限区间上的数据向区间外延拓,但这样做容易产生边界误差,如何构造区间上的具有良好性质的小波,如光滑性、对称性等是非常有意义的．在文献［２］中,Cohen,Darbechies与Vial在总结了构造区间周期小波,折叠小波等方法的基础上,提出了一种新的构造方法,并把无穷区间上的Daubechies小波改造成区间上的Daubechies小波．但是,Daubechies小波没有对称性,光滑性也差．本文用类似的方法把无穷区间上的双正交小波改造…     

联系Butterworth滤波器的双正交小波

联系Butterworth滤波器的双正交小波称为Butterworth小波,它们具有很好的性质:包括对称性,插值性及消失矩．本文定义了离散空间L~2(Z)中的双正交小波并给出一个易于验证的充分条件．利用这一条件,重新得到Butterworth小波;进一步,构造了一类双正交小波．它们不仅具有Butterworth小波的前述所有性质,而且具有最短可能的支集．     

二维周期基数插值小波尺度函数

设f（x）是一个Fourier系数为正的周期函数，我们构造了关f（x）的二维周期基数插值小波的尺度函数，并得到了一些对构造小波函数有重要意义的性质．     

双正交多重小波的一种构造方法

多重小波是近年来新兴的小波研究方向,它具有许多一维小波所不具备的优越性质．完全正交的多重小波在构造上有很大的难度,所以在许多应用中人们都可以用双正交多重小波作为分析的工具     

小波变换及其对厄尔尼诺研究的初步应用

任福民等．小波变换及其对厄尔尼诺研究的初步应用．数理统计与管理，１９９８，１７（３），２１～２５．小波分析是近十多年来迅速成熟起来的具有良好的时域和频域局部分析性质的数学分支，其核心内容是小波变换。本文简要介绍小波变换及相关内容；然后将小波变换用于研究厄尔尼诺（Ｅｌｎｉｎｏ）的时－频特征，结果表明，强度极强的厄尔尼诺只有在几乎所有周期的振荡都表现为较强的正位相时才会发生；１９９７年发生的厄尔尼诺将可能成为一次新的最强的厄尔尼诺事件     

一个具有非局部效应的非线性周期反应扩散方程的渐近形态

研究一个具有非线性-非局部反应的周期反应扩散系统.利用周期半流的渐近理论来讨论渐近波速c~*和周期行波解的存在性,证明参数c~*也是周期行波解的最小波速,并清晰描述解传播的阈值性质.最后给出渐近波速和最小波速c~*的估计.     

小波分析中的一个非线性算子

在这篇文章里,我们以算子的观点考虑了小波的构造问题．结果,我们得到了小波分析中一个非线性算子并调查了这个非线性算子的一些性质．     

Slepian半小波基函数及其在无线通讯信号概率密度估计中的应用

文[20]引进了Slepian半小波基函数并讨论了这组基在概率度估计核方法中的应用[21]，Slepian半小波基函数具有极好的性质．包括多重尺度结构和局部非负性．更值得指出的是．与Gauss核不同，Slepian函数是与无线信号类似的具有平滑谱的有限带宽函数．在所有相同带宽的函数中．Slepian函数在特定的时同区域上具有最大能量．在逼近具有平滑谱的无线信号中．这些特性使得Slepian半小波核与Gauss核以及其他小波基相比具有潜在的优越性．美中不足的是．和其他核密度估计一样．Slepian核密度估计的算法设计具有一定的挑战性．幸运的是．我们注意到Slepian核可以被表示成卷积形式．这一观察具有重要的计算意义．本文主要讨论Slephn核密度估计的应用及其计算．我们首先设计了基于离散卷积的算法并讨论了这一算法的有效性．在文章的结尾，以Slepian核密度估计作为具有平滑谱的远程信号的衰减包络的模型为例．我们考查了Slepian核及其算法的性质．为了尝试数学理论与应用的紧密联系，本文的数值试验不仅采用了模拟数据而且包括了从无线通讯用户的硬件直接采集的实际数据．     

正交小波变换的向量空间算法

本文揭示了一个事实，小波不仅可构成Ｌ２空间中的正交基，小波分解与重构滤波还可产生Ｎ维空间中的正交基．在本文提出修改的小波变换算法之下，Ｎ点信号的小波变换等价于Ｎ维空间中的正交变换．用该算法进行信号或图象压缩，无需对信号或图象进行周期延拓，可严格地在Ｎ维空间中进行．     

Periodic Prolate Spheroidal Wavelets

Prolate spheroidal wavelets (PS wavelets) were recently introduced by the authors. They were based on the first prolate spheroidal wave function (PSWF) and had many desirable properties lacking in other wavelets. In particular, the subspaces belonging to the associated multiresolution analysis (MRA) were shown to be closed under differentiation and translation. In this paper, we introduce periodic prolate spheroidal wavelets. These periodic wavelets are shown to possess properties inherited from PS wavelets such as differentiation and translation. They have the potential for applications in modeling periodic phenomena as an alternative to the usual periodic wavelets as well as the Fourier basis.     

REAL-VALUED PERIODIC WAVELETS: CONSTRUCTION ANDRELATION WITH FOURIER SERIES

1.IntroductionWaveletshaverecentlyreceivedagreatdealofattentioninsuchareasassignalprocessingandimageprocessing([12],[8]).Variousmethodstoconstructwaveletshavebeengiven([14],[13],[9],[7]).Itiswellknownthatinmathematicsandmathematicphysicsmanyperiodicp...     

Construction of Periodic Prolate Spheroidal Wavelets Using Interpolation

Periodic prolate spheroidal wavelets (periodic PS wavelets), based on the periodizaton of the first prolate spheroidal wave function (PSWF), were recently introduced by the authors. Because of localization and other properties, these periodic PS wavelets could serve as an alternative to Fourier series for applications in modeling periodic signals. In this paper, we continue our work with periodic PS wavelets and direct our attention to their construction via interpolation. We show that they have a representation in terms of interpolation with the modified Dirichlet kernel. We then derive a group of formulas of interpolation type based on this representation. These formulas enable one to obtain a simple procedure for the calculation of the periodic PS wavelets and finding expansion coefficients. In particular, they are used to compute filter coefficients for the periodic PS wavelets. This is done for a number of concrete cases.     

Periodic dyadic wavelets and coding of fractal functions

Recently, using the Walsh-Dirichlet type kernels, the first author has defined periodic dyadic wavelets on the positive semiaxis which are similar to the Chui-Mhaskar trigonometric wavelets. In this paper we generalize this construction and give examples of applications of periodic dyadic wavelets for coding the Riemann, Weierstrass, Schwarz, van der Waerden, Hankel, and Takagi fractal functions.     

BIVARIATE REAL-VALUED ORTHOGONAL PERIODIC WAVELETS

In this paper, we construct a kind of bivariate real-valued orthogonal periodic wavelets. The corresponding decomposition and reconstruction algorithms involve only 8 terms respectively which are very simple in practical computation. Moreover, the relation between periodic wavelets and Fourier series is also discussed.     

Two-dimension periodic cardinal interpolatory wavelets

In this paper, we construct two-dimension periodic interpolatory scaling function and wavelets from a periodic function g(x₁, x₂), whose Fourier coefficients are positive, and obtain some properties of scaling functions and wavelets.     

二元实值正交周期Box样条小波

In this paper, we construct a kind of bivariate real-valued orthogonal periodic box-spline wavelets. There are only 4 terms in the two-scale dilation equations. This implies that the corresponding decomposition and reconstruction algorithms involve only 4 terms respectively which are simple in practical computation. The relation between the periodic wavelets and Fourier series is also discussed.     

Two-dimension periodic cardinal interpolatory wavelets

In this paper, we construct two-dimension periodic interpolatory scaling function and wavelets from a periodic function g(x₁, x₂), whose Fourier coefficients are positive, and obtain some properties of scaling functions and wavelets.     

On the detection of singularities of a periodic function

We discuss the problem of detecting the location of discontinuities of derivatives of a periodic function, given either finitely many Fourier coefficients of the function, or the samples of the function at uniform or scattered data points. Using the general theory, we develop a class of trigonometric polynomial frames suitable for this purpose. Our methods also help us to analyze the capabilities of periodic spline wavelets, trigonometric polynomial wavelets, and some of the classical summability methods in the theory of Fourier series. This revised version was published online in June 2006 with corrections to the Cover Date.     

The use of discrete dyadic wavelets in image processing

In this paper, using the discreteWalsh transform, we construct orthogonal and biorthogonal wavelets for complex periodic sequences similar to those studied earlier for the Cantor group. Results of numerical experiments demonstrate the effectiveness of the use of constructed discrete wavelets in image processing.