A trade-off principle in connection with the approximation by positive definite kernels
نویسنده
چکیده
A frequently used method to handle scattered data approximation problems on different structures is the interpolation by linear combinations of a single positive definite kernel. Here the underlying positive definite kernel is responsible for the quality of the common estimates of the two main issues in this approximation process. Firstly the approximation error and secondly the stability of the system. In this paper it is shown that we can not ask the kernel for arbitrary good estimates for the attainable error and the condition number of the associated interpolation matrix at the same time. Hereby we use a generalization of Schaback’s trade-off principle for RBF interpolation in R [R. Schaback. Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math., 3, 251–264, 1995.] to more general settings and a wider class of basis functions. §
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