曲面样条逼近中的多调和狄利克雷问题及边界效应
On a polyharmonic Dirichlet problem and boundary effects in surface spline approximation
摘要：
对于具有光滑边界的紧致域，我们提出了一个曲面样条逼近的逼近方案，该方案在已知的光滑空间上提供精确的$L_p$逼近阶。该格式克服了中心在边界附近密度较大时的边界效应。它的成功归功于使用最少数量的边界层势的积分恒等式，而边界层势又是从$m$ -折叠拉普拉斯算子的狄利克雷问题的边界层势解导出的。此外，他的积分恒等式被证明是目标函数的“本机空间扩展”。
英文摘要：
For compact domains with smooth boundaries, we present an approximation scheme for surface spline approximation that delivers precise $L_p$ approximation orders on well known smoothness spaces. This scheme overcomes the boundary effects when centers are placed with greater density near to the boundary. It owes its success to an integral identity using a minimal number of boundary layer potentials, which, in turn is derived from the boundary layer potential solution to the Dirichlet problem for the $m$-fold Laplacian. Furthermore, his integral identity is shown to be the "native space extension" of the target function.
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