Compactly supported tight frames associated with refinable functions
Compactly supported tight frames associated with refinable functions
复制标题
DOI：
10.1006/acha.2000.0301
复制
发表时间：
2000
期刊：
影响因子：
--
通讯作者：
C. Chui;Wenjie He
中科院分区：
文献类型：
--
作者：
C. Chui;Wenjie He
在应用数学和计算数学中，基数B样条在几何建模（计算机辅助几何设计）、统计数据表示（或建模）、微分方程求解（数值分析）等方面起着重要作用。最近，在小波分析的发展中，基数B样条也作为产生L2（−∞，∞）的多分辨率分析的尺度函数的典型例子。然而，虽然基数B样条有紧支撑，但它们对应的标准正交小波（Battle和Lemarie）有无限的持续时间。为了保持自对偶等性质，同时要求紧的支持，紧框架的概念可能是唯一的替代正交小波。本文研究了L2（-∞，∞）上对应于某些具有紧支撑的可加细函数的紧支撑紧框架Ψ={ψ1，.，ψN}，根据可加细函数的洛朗多项式符号上的不等式条件给出了Ψ的精确存在性准则，表明该条件并不总是满足（通过矩阵延拓方法证明了紧框架的不存在性），并给出了一个构造性的证明，证明了当紧框架存在时，两个具有紧支集的函数足以构成紧框架，而三个函数保证对称/反对称，当给定的可加细函数是对称的。
Abstract It is well known that in applied and computational mathematics, cardinal B-splines play an important role in geometric modeling (in computer-aided geometric design), statistical data representation (or modeling), solution of differential equations (in numerical analysis), and so forth. More recently, in the development of wavelet analysis, cardinal B-splines also serve as a canonical example of scaling functions that generate multiresolution analyses of L2(−∞,∞). However, although cardinal B-splines have compact support, their corresponding orthonormal wavelets (of Battle and Lemarie) have infinite duration. To preserve such properties as self-duality while requiring compact support, the notion of tight frames is probably the only replacement of that of orthonormal wavelets. In this paper, we study compactly supported tight frames Ψ={ψ1,…,ψN} for L2(−∞,∞) that correspond to some refinable functions with compact support, give a precise existence criterion of Ψ in terms of an inequality condition on the Laurent polynomial symbols of the refinable functions, show that this condition is not always satisfied (implying the nonexistence of tight frames via the matrix extension approach), and give a constructive proof that when Ψ does exist, two functions with compact support are sufficient to constitute Ψ, while three guarantee symmetry/anti-symmetry, when the given refinable function is symmetric.