一个二进小波变换像空间的再生核函数  

THE REPRODUCING KERNEL FUNCTION OF IMAGE SPACE OF A DYADIC WAVELET TRANSFORM

作　　者：王翠 邓彩霞 刘国宁 陈夏夏 Wang Cui;Deng Caixia;Liu Guoning;Chen Xiaxia(Department of Mathematics,School of Science,Harbin University of Science and Technology,Harbin 150080;Department of Electrical Engineering,School of Electrical and Electronic Engineering,Harbin University of Science and Technology,Harbin 150080;Phoenix High School,Heze 274000)

机构地区：[1]哈尔滨理工大学理学院数学系,哈尔滨150080 [2]哈尔滨理工大学电气与电子工程学院电气工程系,哈尔滨150080 [3]凤凰中学,菏泽274000

出　　处：《高等学校计算数学学报》2021年第1期1-15,共15页Numerical Mathematics A Journal of Chinese Universities

基　　金：国家自然科学基金(11871181).

摘　　要：1 引言小波分析是结合泛函分析、应用数学、逼近论、调和分析、广义函数论等数学知识的结晶,具有深刻的理论意义和广泛的应用范围,被称为"数学显微镜".基于其多分辨分析的特点以及在时、频两域都具有表征信号局部特征的功能,应用它可以解决许多Fourier变换不能解决的难题,为工程应用提供了一种新的、更有效的分析工具[1].Firstly,this paper constructs a dyadic wavelet with non-orthogonality,symmetry,limited spectrum and full smooth almost everywhere,discusses the dual wavelet and its properties,and gets the isometric and the estimation of the dyadic wavelet transform.Secondly,the analytic expression of the re-producing kernel function of image space of the dyadic wavelet transform is obtained by using the reproducing kernel Hilbert space theory.When the scale factor is fixed,the reproducing kernel function of image space of the dyadic wavelet transform is further characterized.Reproducing kernel illustrates the redundancy of wavelet transform so that the dyadic wavelet transform can be represented by reproducing kernel.Finally different forms of expansion of dyadic wavelet transform are given,and the inverse formula of dyadic wavelet transform are obtained by using dyadic dual wavelet.The proposed theorems not only provide a reference for the study of image space of general wavelet transform,but also provide a theoretical basis for the numerical calculation of wavelet in engineering applications.

关 键 词：多分辨分析 泛函分析 广义函数论 应用数学 调和分析 逼近论 小波分析 二进小波变换 

分 类 号：O175.14[理学—数学]