单个生成元Walsh p-进制平移不变空间伸缩的交与并  

The Intersection and Union of Dilates of Singly Generated Walsh p-adic Shift-invariant Spaces

作　　者：张岩 李云章[2] Yan ZHANG;Yun Zhang LI(School of Malhematics and Information Science,North Minzu University, Yinchuan 750021,P.R.China;College of Applied Sciences,Beijing University of Technology, Beijing 100124,P.R.China)

机构地区：[1]北方民族大学数学与信息科学学院,银川750021 [2]北京工业大学应用数理学院,北京100124

出　　处：《数学学报（中文版）》2019年第1期1-12,共12页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助课题(11501010;11271037);宁夏高等学校科学研究项目(NGY2018-163)

摘　　要：p-进制MRA与GMRA是构造L^2(R_+)中小波框架的重要工具. L^2(R+)中嵌套子空间序列交集为{0},并集为L^2(R_+)是其构成p-进制MRA与GMRA的基本要求.本文研究单个生成元Walsh p-进制平移不变子空间伸缩的交与并,证明了:对任意单个生成元Walsh p-进制平移不变子空间,其p-进制伸缩的交是{0};若生成元分为Walsh p-细分函数,则其p-进制伸缩的并是L^2(R_+)中一个Walshp-进制约化子空间.特别地,其伸缩构成L^2(R_+)中p-进制GMRA当且仅当∪_(j∈z)p^j supp(■φ)=R+,其中■为定义在L^2(R_+)上的Walsh p-进制傅里叶变换.值得注意的是:形式上,我们的结果类似于通常L^2(R)的情形,然而其证明不是平凡的.这是因为定义在R_+上的p-进制加法"⊕"不同于定义在R上的通常加法"+".p-adic MRA and GMRA are important tools for constructing wavelet frames in L^2(R+). That a nested subspace sequence in L^2(R+) has trivial intersection and L^2(R+) union is a fundamental requirement for it to form a p-adic MRA and GMRA. This paper addresses the intersection and union of p-adic dilates of a singly generated p-adic shift-invariant subspace. We prove that, for a singly generated p-adic shift-invariant subspace, the intersection of its p-adic dilates is {0}, and the union of its p-adic dilates is a Walsh p-adic reducing subspace of L^2(R+) if the generator φ is Walsh p-adic refinable in addition. In particular, the dilates form a p-adic GMRA for L^2(R+) if and only if ∪j∈zpj supp(■.φ)=R+,where ■ is the Walsh p-adic Fourier transform on L^2(R+). It is worth noticing that our results are similar to the case of usual L^2(R),while their proofs are nontrivial. It is because the p-adic addition ⊕ on R+ is different from the usual addition + on R.

关 键 词：框架 p-进制小波框架 WALSH p-进制细分函数 

分 类 号：O174.2[理学—数学]