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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
p-进制MRA与GMRA是构造L2(R+)中小波框架的重要工具.L2(R+)中嵌套子空间序列交集为0，并集为L2(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φ为Walsh p-细分函数，则其p-进制伸缩的并是L2(R+)中一个Walsh p-进制约化子空间.特别地，其伸缩构成L2(R+)中p-进制GMRA当且仅当∪j∈Zpjsupp（Fφ）=R+，其中F为定义在L2(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L2（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 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∈Zpjsupp(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.
框架 / p-进制小波框架 / Walsh p-进制细分函数 {{custom_keyword}} /
frame / p-adic wavelet frame / Walsh p-adic refinable function {{custom_keyword}} /
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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）
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