论文标题
源自任意紧凑型生物双管多波武器的间隔小波
Wavelets on Intervals Derived from Arbitrary Compactly Supported Biorthogonal Multiwavelets
论文作者
论文摘要
（BI）真实线上的正交（多）小波已在成功的应用中进行了广泛的研究和使用。应用程序中的许多问题是在有限的间隔或域上定义的。因此，在理论和应用中都重要的是，在实际线上具有（BI）正交（多）小波的某些所需属性的间隔构建所有可能的小波。消失的紧凑型小波的力矩是稀疏小波表示的关键特性，并且与其潜在可再连接（向量）函数的多项式复制密切相关。具有低阶消失力矩的边界小波通常会导致不希望的边界伪像以及稀疏性和近似顺序的降低。 From any arbitrarily given compactly supported (bi)orthogonal multiwavelet on the real line, in this paper we propose two different approaches to construct/derive all possible locally supported (bi)orthogonal (multi)wavelets on $[0,\infty)$ or $[0,1]$ with or without prescribed vanishing moments, polynomial reproduction, and/or homogeneous boundary conditions.第一种方法将经典方法从标量小波到多波器进行了概括，而第二种方法是直接的，而无需明确涉及任何双重修补功能和双重多波武器。 （多）在满足同质边界条件的间隔的小波中也将被解决。尽管在间隔上构建正交（多）小波要比其生物双歧架对应物要容易得多，但我们表明，如果这些正交（多）小波在间隔上满足同质性的dirichlet边界条件，则某些边界正交小波不会具有任何消失的矩。将提供几个在间隔$ [0,1] $上的正交和生物多的多波武器的示例，以说明我们的施工方法和建议的算法。
(Bi)orthogonal (multi)wavelets on the real line have been extensively studied and employed in applications with success. A lot of problems in applications are defined on bounded intervals or domains. Therefore, it is important in both theory and application to construct all possible wavelets on intervals with some desired properties from (bi)orthogonal (multi)wavelets on the real line. Vanishing moments of compactly supported wavelets are the key property for sparse wavelet representations and are closely linked to polynomial reproduction of their underlying refinable (vector) functions. Boundary wavelets with low order vanishing moments often lead to undesired boundary artifacts as well as reduced sparsity and approximation orders near boundaries in applications. From any arbitrarily given compactly supported (bi)orthogonal multiwavelet on the real line, in this paper we propose two different approaches to construct/derive all possible locally supported (bi)orthogonal (multi)wavelets on $[0,\infty)$ or $[0,1]$ with or without prescribed vanishing moments, polynomial reproduction, and/or homogeneous boundary conditions. The first approach generalizes the classical approach from scalar wavelets to multiwavelets, while the second approach is direct without explicitly involving any dual refinable functions and dual multiwavelets. (Multi)wavelets on intervals satisfying homogeneous boundary conditions will also be addressed. Though constructing orthogonal (multi)wavelets on intervals is much easier than their biorthogonal counterparts, we show that some boundary orthogonal wavelets cannot have any vanishing moments if these orthogonal (multi)wavelets on intervals satisfy the homogeneous Dirichlet boundary condition. Several examples of orthogonal and biorthogonal multiwavelets on the interval $[0,1]$ will be provided to illustrate our construction approaches and proposed algorithms.
