任意矩阵伸缩的双向正交小波包

基于双向加细小波函数和双向小波尺度函数,给出了矩阵伸缩的双向小波的定义;给出矩阵伸缩的多分辨分析,并给出矩阵伸缩的小波的正交条件,得到矩阵伸缩的双向小波包;并得到相关性质和结论.     

用符号刻划多变量加细函数向量的精度

假定ｆ（ｘ）＝（ｆ１（ｘ）,…,ｆｒ（ｘ））Ｔ,ｘ∈Ｒｄ是一个满足矩阵加细方程ｆ（ｘ）＝的向量函数,其中Ａ是Ｚｄ中的有限集,ｃｋ是ｒ×ｒ,矩阵．由小波分析的多尺度分析方法可知,加细向量ｆ的性质需要由符号叫ｍ（ξ）来反映．本文,我们给出了用符号来刻划多变量加细向量ｆ具有Ｐ阶精度的充分必要条件．     

对称的高逼近阶多小波的构造

本文基于已有的对称多小波,给出构造对称的高逼近阶多小波的一个显式算法．具体地,假设Φ（x）：=（φ1（x）…．,φr（x））T是一个具有逼近阶m的对称加细函数向量．对于任意非负整数n,一个具有逼近阶m＋n的新对称加细函数向量Φ^new（x）：=（φ1^new（x）…．,φr^new（x））^T可由上述算法构造出来．另外,揭示了Φ（x）与Φ^new（x）之间的关系．为了使我们的结果具体化,从具有逼近阶4的三次Hermite函数出发,构造了一个具有逼近阶6的对称加细函数向量．     

a尺度三维八向加细小波尺度函数的构造算法

依据双向小波理论为基础,利用张量积构造a尺度三维八向加细小波尺度函数,研究了具有紧支撑性面具符号的a尺度三维八向加细方程拥有紧支撑分布解的情况,讨论了三维八向a尺度加细小波函数紧支撑区间.     

周期多尺度分析的特征及它的一个应用

在这篇文章里，我们研究了周期多尺度分析的性质，给出了尺度函数序列的一个特征．这个特征能够使我们从一个尺度函数序列得到另一个尺度函数序列．最后，我们给出了主要结果的一个应用．     

L2(Rs)中插值小波包的分裂技巧和精细分解

对具有任意伸缩矩阵A的插值加细函数，给出对应于L^2(R^s)中的小波包的一个构造方法．采样空间被直接分解来取代对加细函数的符号分解．按照这个方法构造的插值小波包能对基插值空间提供较为精细的分解，因而对自适应的插值给出较好的局部化．     

二维连续信号的近似采样定理

利用加细方程的面具,给出该方程的一个近似解,并根据这个近似解构造出二维连续信号的近似采样定理.其近似采样函数是所求加细方程的近似解,它是由加细方程的面具唯一确定的逐段线性函数,且有显示的计算公式.因此可以根据需要选择加细方程的面具,从而达到控制近似采样函数的衰减速度.     

a尺度正交多尺度函数和正交多小波

基于a 尺度正交单尺度函数，分别给出重数为2和3的a 尺度正交多尺度函数的构造算法。并给出对应正交多小波的显式构造。最后给出伸缩因子为3的正交多小波的构造算例。     

最小能量多重仿射紧框架的特征

引入多个加细函数生成的最小能量多重仿射框架的概念.运用矩阵理论和时频分析方法,给出对应于最小能量多重仿射框架的多重加细函数所满足的条件.得到最小能量多重仿射紧框架的特征与构造方法.对原多重加细函数和最小能量多小波仿射框架做正交变换,得到新的多重加细函数和对应的最小能量多重仿射框架.     

区间[-1，1]上的α尺度双正交多小波的构造

本文研究了一元α尺度紧支撑、双正交多小波的构造．在区间[-1，1]，给出了利用α尺度双正交尺度向量构造α尺度双正交多小波的推导过程得到了一种有效的小波构造算法，并给出了数值算例．     

Compactly supported (bi)orthogonal wavelets generated by interpolatory refinable functions

This paper provides several constructions of compactly supported wavelets generated by interpolatory refinable functions. It was shown in [7] that there is no real compactly supported orthonormal symmetric dyadic refinable function, except the trivial case; and also shown in [10,18] that there is no compactly supported interpolatory orthonormal dyadic refinable function. Hence, for the dyadic dilation case, compactly supported wavelets generated by interpolatory refinable functions have to be biorthogonal wavelets. The key step to construct the biorthogonal wavelets is to construct a compactly supported dual function for a given interpolatory refinable function. We provide two explicit iterative constructions of such dual functions with desired regularity. When the dilation factors are larger than 3, we provide several examples of compactly supported interpolatory orthonormal symmetric refinable functions from a general method. This leads to several examples of orthogonal symmetric (anti‐symmetric) wavelets generated by interpolatory refinable functions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Refinement equations and spline functions

In this paper, we exploit the relation between the regularity of refinable functions with non-integer dilations and the distribution of powers of a fixed number modulo 1, and show the nonexistence of a non-trivial C ^∞ solution of the refinement equation with non-integer dilations. Using this, we extend the results on the refinable splines with non-integer dilations and construct a counterexample to some conjecture concerning the refinable splines with non-integer dilations. Finally, we study the box splines satisfying the refinement equation with non-integer dilation and translations. Our study involves techniques from number theory and harmonic analysis.     

Local Regularity of L∞ ∞-Refinable Function Vectors

This article discusses the local regularity of refinable function vectors associated with a dilation matrix M. Suppose that D is a complete set of representatives of Z^s/MZ^s. Under the assumptions that the self-affine tile T (M,D), associated with dilation matrix M and digit set D, has measure 1 and that the corresponding refinable function vector φ ∈ L_∞, we prove that there is a set H ? R^s of full measure such that the restriction φ|_H of φ on H has a positive Hölder exponent α(x) at every x ∈ H. Similar result holds for the derivatives of refinable function vectors provided that the dilation matrix is diagonalizable.     

On Multivariate Compactly Supported Bi-Frames

In this article, we construct compactly supported multivariate pairs of dual wavelet frames, shortly called bi-frames, for an arbitrary dilation matrix. Our construction is based on the mixed oblique extension principle, and it provides bi-frames with few wavelets. In the examples, we obtain optimal bi-frames, i.e., primal and dual wavelets are constructed from a single fundamental refinable function, whose mask size is minimal w.r.t. sum rule order and smoothness. Moreover, the wavelets reach the maximal approximation orderw.r.t. the underlying refinable function. For special dilation matrices, we derive very simple but optimal arbitrarily smooth bi-frames in arbitrary dimensions with only two primal and dual wavelets.     

Refinable functions with non-integer dilations

Refinable functions and distributions with integer dilations have been studied extensively since the pioneer work of Daubechies on wavelets. However, very little is known about refinable functions and distributions with non-integer dilations, particularly concerning its regularity. In this paper we study the decay of the Fourier transform of refinable functions and distributions. We prove that uniform decay can be achieved for any dilation. This leads to the existence of refinable functions that can be made arbitrarily smooth for any given dilation factor. We exploit the connection between algebraic properties of dilation factors and the regularity of refinable functions and distributions. Our work can be viewed as a continuation of the work of Erdös [P. Erdös, On the smoothness properties of a family of Bernoulli convolutions, Amer. J. Math. 62 (1940) 180-186], Kahane [J.-P. Kahane, Sur la distribution de certaines séries aléatoires, in: Colloque de Théorie des Nombres, Univ. Bordeaux, Bordeaux, 1969, Mém. Soc. Math. France 25 (1971) 119-122 (in French)] and Solomyak [B. Solomyak, On the random series ∑±λn (an Erdös problem), Ann. of Math. (2) 142 (1995) 611-625] on Bernoulli convolutions. We also construct explicitly a class of refinable functions whose dilation factors are certain algebraic numbers, and whose Fourier transforms have uniform decay. This extends a classical result of Garsia [A.M. Garsia, Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc. 102 (1962) 409-432].     

Local dual and poly-scale refinability

For a compactly supported function , let , be the space of all finite linear combinations of . In this paper, we consider the explicit construction of local duals of and the poly-scale refinability of functions in when is an -refinable function. We show that for any -refinable function , there exists a local dual of in for some , and that any function in is poly-scale refinable.

     

Smoothness of multivariate refinable functions with infinitely supported masks

In this paper, we investigate the smoothness of multivariate refinable functions with infinitely supported masks and an isotropic dilation matrix. Using some methods as in [R.Q. Jia, Characterization of smoothness of multivariate refinable functions in Sobolev spaces, Trans. Amer. Math. Soc. 351 (1999) 4089–4112], we characterize the optimal smoothness of multivariate refinable functions with polynomially decaying masks and an isotropic dilation matrix. Our characterizations extend some of the main results of the above mentioned paper with finitely supported masks to the case in which masks are infinitely supported.     

A NOTE ON VECTOR CASCADE ALGORITHM

AbstractThe focus of this paper is on the relationship between accuracy of multivariate refinable vector and vector cascade algorithm. We show that, if the vector cascade algorithm (1.5) with isotropic dilation converges to a vector-valued function with regularity, then the initial function must satisfy the Strang-Fix conditions.     

The support of a refinable vector satisfying an inhomogeneous refinement equation

In this paper, we investigate the support of a refinable vector satisfying an inhomoge- neous refinement equation. By using some methods introduced by So and Wang, an estimate is given for the support of each component function of a compactly supported refinable vector satisfying an inhomogeneous matrix refinement equation with finitely supported masks.