Positive-definite kernels are probably best known for their application in many problems driven by scattered data interpolation. Fasshauer and Ye introduced constructive theory of reproducing kernels of generalized Sobolev spaces in 2011 to provide insight into the types of functions being well approximated by these kernels on a set of scattered points. In this approach, the reproducing kernel is viewed as the Green kernel of a suitable differential operator with some boundary conditions. Sampling inequalities and the minimum norm property in reproducing kernel Hilbert spaces (RKHSs) bring out the standard error bound; however, this estimate is valid only when the target functions belong to the native spaces of the Green kernels. In this paper we provide Sobolev-type error estimates for cases in which the target functions are smoother than functions in the native space. The results are useful and effective for the error analysis of Green kernel-based interpolation problems.

正定核可能以其在由分散数据插值驱动的许多问题中的应用而闻名。Fasshauer 和 Ye 在 2011 年引入了广义 Sobolev 空间核再现的建设性理论，以深入了解这些核在一组散点上很好地逼近的函数类型。在这种方法中，再生核被视为具有一些边界条件的合适微分算子的格林核。再现核希尔伯特空间 (RKHS) 中的抽样不等式和最小范数性质带出标准误差界；然而，这个估计只有在目标函数属于格林内核的原生空间时才有效。在本文中，我们为目标函数比原生空间中的函数更平滑的情况提供了 Sobolev 类型的误差估计。结果对于基于格林核的插值问题的误差分析是有用和有效的。