单个生成元Walsh p-进制平移不变空间伸缩的交与并

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"+".

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

p-adic MRA and GMRA are important tools for constructing wavelet frames in L²(R₊). That a nested subspace sequence in L²(R₊) has trivial intersection and L²(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 L²(R₊) if the generator φ is Walsh p-adic refinable in addition. In particular, the dilates form a p-adic GMRA for L²(R₊) if and only if ∪_j∈Zp^jsupp(Fφ)=R₊, where F is the Walsh p-adic Fourier transform on L²(R₊). It is worth noticing that our results are similar to the case of usual L²(R), while their proofs are nontrivial. It is because the p-adic addition ⊕ on R₊ is different from the usual addition + on R.