The characterization of low pass filters and some basic properties of wavelets, scaling functions and related concepts
The characterization of low pass filters and some basic properties of wavelets, scaling functions and related concepts
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低通滤波器的表征以及小波的一些基本性质、缩放函数和相关概念
DOI：
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发表时间：
1999
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通讯作者：
G. Weiss
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文献类型：
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作者：
M. Papadakis;H. Šikić;G. Weiss
摘要“经典”小波，即 ψ εL2(R)，使得 {2j/2ψ(2jx−k)}, j,kεZ 是 L2 (R) 的正交基，已知其特征在于两个简单方程满足：
$$帽子 psi $$
。 “多分辨率分析”小波（简称 MRA 小波）具有简单的表征，产生这些小波的缩放函数也是如此。然而，只有由这些缩放函数确定的低通滤波器的某些平滑类别在文献中出现了特征（有关这些问题的说明，请参阅 [3] 的第 7 章）。在本文中，我们展示了所有这些滤波器的完整特征。这个有点技术性的成果确实提供了一种简单构造低通滤波器的方法，其唯一的平滑度假设是原点处的 Holder 条件。我们还获得了所有缩放集的表征，特别是所有有界缩放集的描述以及缩放函数类别的详细描述。
AbstractThe “classical” wavelets, those ψ εL2(R) such that {2j/2ψ(2jx−k)}, j,kεZ, is an orthonormal basis for L2 (R), are known to be characterized by two simple equations satisfied by
$$hat psi $$
. The “multiresolution analysis” wavelets (briefly, the MRA wavelets) have a simple characterization and so do the scaling functions that produce these wavelets. Only certain smooth classes of the low pass filters that are determined by these scaling functions, however, appear to be characterized in the literature (see Chapter 7 of [3] for an account of these matters). In this paper we present a complete characterization of all these filters. This somewhat technical result does provide a method for simple constructions of low pass filters whose only smoothness assumption is a Holder condition at the origin. We also obtain a characterization of all scaling sets and, in particular, a description of all bounded scaling sets as well as a detailed description of the class of scaling functions.