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Derivative-orthogonal Riesz wavelets in Sobolev spaces with applications to differential equations

机译:Sobolev空间中的导数正交Riesz小波及其在微分方程中的应用

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Riesz wavelets in the Sobolev space H-m(R) with m is an element of N boolean OR {0}, whose mth-order derivatives are orthogonal among different levels, are of particular interest and importance in computational mathematics, due to their many desirable properties such as small condition numbers and sparse stiffness matrices. We call such Riesz wavelets in the Sobolev space H-m(R) as mth-order derivative-orthogonal Riesz wavelets. In this paper we shall comprehensively study and completely characterize all compactly supported mth-order derivative-orthogonal Riesz wavelets in the Sobolev space H-m(R). More precisely, from any given compactly supported refinable vector function phi = (phi(1), ..., phi(r))(T) in H-m(R) satisfying the refinement equation (phi) over cap (2 xi) = (a) over cap(xi) (phi) over cap(xi) for some r x r matrix (a) over cap of 2 pi-periodic trigonometric polynomials, we prove that there exists a compactly supported mth-order derivative-orthogonal Riesz wavelet in H-m (R), which is derived from phi through the refinable structure, if and only if the refinable vector function phi has stable integer shifts and the filter a has at least order 2m sum rules. This double order of sum rules over the smoothness order m is surprising but is necessary for constructing mth-order derivative-orthogonal Riesz wavelets in H-m(R). Then we shall present several examples of such derivative-orthogonal spline Riesz wavelets with short support derived from B-splines and Hermite splines. To illustrate the developed theory and its potential usefulness, we shall apply our constructed such mth-order derivative-orthogonal Riesz wavelets for the numerical solutions of differential equations such as Sturm-Liouville equations and biharmonic equations. Our constructed derivative-orthogonal spline Riesz wavelets on the interval [0, 1] have a simple structure with only one boundary wavelet at each endpoint and can easily handle different types of boundary conditions. The resulting coefficient matrices are sparse and have very small condition numbers with some examples even having the optimal condition number one. (C) 2017 Elsevier Inc. All rights reserved.
机译:具有m的Sobolev空间Hm(R)中的Riesz小波是N布尔OR {0}的元素,其m阶导数在不同水平之间是正交的,由于它们具有许多理想的特性,因此在计算数学中尤为重要例如较小的条件数和稀疏的刚度矩阵。我们将Sobolev空间H-m(R)中的此类Riesz小波称为m阶导数-正交Riesz小波。在本文中,我们将全面研究和完全表征Sobolev空间H-m(R)中所有紧支持的m阶导数-正交Riesz小波。更精确地讲,从任何给定的紧凑支持的可精炼向量函数phi =(phi(1),...,phi(r))(T)在Hm(R)中满足上限(2 xi)= (a)对于某些rxr矩阵的cap(xi)(phi)over cap(xi)(a)在2个pi周期三角多项式的cap之上,我们证明存在一个紧致支持的m次阶导数-正交Riesz小波Hm(R),当且仅当可精炼矢量函数phi具有稳定的整数移位并且滤波器a具有至少2m阶求和规则时,才通过可精炼结构从phi导出Hm(R)。这种在平滑度阶m上的和规则的双阶令人惊讶,但对于在H-m(R)中构造m阶导数-正交Riesz小波是必需的。然后,我们将介绍从B样条和Hermite样条获得的短支持的此类导数正交样条Riesz小波的几个示例。为了说明已发展的理论及其潜在的实用性,我们将构造的m阶导数-正交Riesz小波应用于微分方程(例如Sturm-Liouville方程和双调和方程)的数值解。我们在区间[0,1]上构造的导数-正交样条Riesz小波具有简单的结构,在每个端点只有一个边界小波,并且可以轻松处理不同类型的边界条件。所得系数矩阵稀疏并且具有非常小的条件数,有些示例甚至具有最佳条件数一。 (C)2017 Elsevier Inc.保留所有权利。

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