Some Notes on Error Analysis for Kernel Based Regularized Interpolation

Kernel based regularized interpolation is one of the most important methods for approximating functions. The theory behind the kernel based regularized interpolation is the well-known Representer Theorem, which shows the form of approximation function in the reproducing kernel Hilbert spaces. Because of the advantages of the kernel based regularized interpolation, it is widely used in many mathematical and engineering applications, for example, dimension reduction and dimension estimation. However, the performance of the approximation is not fully understood from the theoretical perspective. In other word, the error analysis for the kernel based regularized interpolation is lacking. In this paper, some error bounds in terms of the reproducing kernel Hilbert space norm and Sobolev space norm are given to understand the behavior of the approximation function