摘要
V-系统是一类有详细数学表达的多小波,它的构造依赖于尺度函数和小波函数.在现有文献中,V-系统的小波函数是通过待定系数法并求解一个非线性方程组得到的.本文用一个全新的方法来构造V-系统的小波函数,基本步骤为:从Legendre多项式和截断单项式出发,通过Gram-Schmidt正交化过程,用递归的方式得到L^2[-1,1]空间的正交函数组,并证明了这个函数组中的部分函数恰是V-系统的小波函数.与现有文献相比,这样得到的任意k次V-小波函数都有明确的数学表达,小波函数的奇偶性非常明确;且任意k次小波函数的存在性得到证明;特别是还给出了高次和低次V-系统的小波函数之间的数学关系.这些工作使得V-系统的数学结构更加清晰,对进一步分析V-系统的数学性质有着重要的理论意义.本文最后给出了一个刻画V-系统特性的简单应用例子,这个例子说明V-系统在分离的群组对象的表达方面,较经典小波更有优势.
The V-system is a special multiwavelet with detailed mathematical expressions. The construction of the V-system is based on the scaling functions and wavelet functions. In the existing literature, the wavelet functions of the V-system are determined by employing the method of undetermined coefficients and thus solving a system of nonlinear equations. This paper uses a new method completely different from that in existing literature to construct the V-system. The basic steps are as follows： Starting from the Legendre polynomial and the truncated monomials, through the Gram-Schmidt orthogonalization process, an orthogonal function group is obtained in L^2[-1, 1] with a recursive method, and some functions in the orthogonal function group are proved to be just the wavelet functions of V-system. The V-wavelet functions obtained in this paper have explicit mathematical structure. The existence of the wavelet functions is also verified, and the relationship between wavelet functions of lower degree and that of higher degree is established, which makes the mathematical structure of the V-system more concisely and has important theoretical significance for further analyzing the mathematical properties of the V-system. An applicable example reflecting the advantages of the V-system is provided in the end of this paper.
出处
《中国科学：数学》
CSCD
北大核心
2016年第6期867-876,共10页
Scientia Sinica：Mathematica
基金
国家重点基础研究发展计划(批准号:2011CB302400)
国家自然科学基金(批准号:61272026)
北京市自然基金重点项目暨北京市教委科技发展计划重点项目(批准号:KZ201210009011)资助项目
关键词
多小波
小波函数
U-系统
V-系统
正交多项式
multiwavelets
wavelet functions
U-system
V-system
orthogonal polynomials
