分数阶Sobolev半范数使用任意阶数据的采样不等式
A sampling inequality for fractional order Sobolev semi-norms using arbitrary order data
AI总结：
通过扩展Arcangéli等人的Sobolev界，纳入任意阶导数的离散样本并优化分数阶Sobolev半范数，以提高非对称无网格方法的收敛性。
摘要：
为了改进使用Schaback（Preprint Göttingen 2006）的非对称无网格方法框架获得的收敛结果，我们在两个方向上扩展了Arcangéli等人（Numer Math 107, 181-211, 2007）的Sobolev界，该界本身扩展了Wendland和Rieger（Numer Math 101, 643-662, 2005）以及Madych（J. Approx Theory 142, 116-128, 2006）的两个结果。第一个方向是将任意阶导数的离散样本纳入界中，用于在高阶Sobolev范数中获得更高阶的收敛性。第二个方向是最优地界定分数阶Sobolev半范数，用于在解决需要分数阶Sobolev空间的问题（特别是非齐次边值问题）时获得更优的收敛速率。
英文摘要：
To improve convergence results obtained using a framework for unsymmetric meshless methods due to Schaback (Preprint Göttingen 2006), we extend, in two directions, the Sobolev bound due to Arcangéli et al. (Numer Math 107, 181-211, 2007), which itself extends two others due to Wendland and Rieger (Numer Math 101, 643-662, 2005) and Madych (J. Approx Theory 142, 116-128, 2006). The first is to incorporate discrete samples of arbitrary order derivatives into the bound, which are used to obtain higher order convergence in higher order Sobolev norms. The second is to optimally bound fractional order Sobolev semi-norms, which are used to obtain more optimal convergence rates when solving problems requiring fractional order Sobolev spaces, notably inhomogeneous boundary value problems.
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