Criteria for Hierarchical Bases in Sobolev Spaces
Criteria for Hierarchical Bases in Sobolev Spaces
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DOI：
10.1006/acha.2000.0275
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发表时间：
2000
影响因子：
2.5
通讯作者：
R. Lorentz;P. Oswald
中科院分区：
文献类型：
--
作者：
R. Lorentz;P. Oswald
数值求解椭圆问题的几种方法都是基于 Sobolev 空间中的分层 Riesz 基。我们感兴趣的是确定 Sobolev 指数的精确范围，从多分辨率分析导出的紧支持函数系统形成了这样的 Riesz 基础。这涉及到确定双系统的平滑度。对偶系统的元素通常由非紧支持函数组成，其平滑度可以通过扩展 7、9 和 22 的结果来处理。我们展示了如何从理论和数值上根据传递算子的谱特性确定多变量情况下 Sobolev 指数的精确范围。该技术适用于文献中提出的从线性有限元导出的几种基础。对于 29 层次基，我们发现它在 Hs(Rd) 中形成了 Riesz 基，为 −0.990236…
Several approaches to solving elliptic problems numerically are based on hierarchical Riesz bases in Sobolev spaces. We are interested in determining the exact range of Sobolev exponents for which a system of compactly supported functions derived from a multiresolution analysis forms such a Riesz basis. This involves determining the smoothness of the dual system. The elements of the dual system typically consist of noncompactly supported functions, whose smoothness can be treated by extending the results of 7, 9, and 22. We show how to determine the exact range of Sobolev exponents in the multivariate case, both theoretically and numerically, from spectral properties of transfer operators. This technique is applied to several bases deriving from linear finite elements which have been proposed in the literature. For 29hierarchical basis, we find that it forms a Riesz basis in Hs(Rd) for −0.990236…