Spectral Approximation Orders of Local Interpolation by Using Radial Basis Functions

The local radial basis function (RBF) interpolation method enables very large-scale data sets to be handled efficiently, overcoming the drawbacks of global interpolation which produces highly ill-conditioned linear systems. Whereas there have been intensive studies on the accuracy of global RBF interpolation, the error analysis of local RBF interpolation is much less investigated. In this regard, this study explores the approximation order of local RBF interpolation. We are particularly interested in using smooth RBFs, including Gaussian functions, because they can provide spectral approximation orders. In fact, most of the previous studies of smooth RBF interpolation have estimated errors for functions in a certain reproducing kernel Hilbert space Fφ. However, since the space Fφ is very small when the function φ is smooth (e.g., a Gaussian function), this study focuses on proving approximation orders of local RBF interpolation for functions in the space C. AMS (MOS) Subject Classification. 41A05, 41A10, 41A25, 41A30, 41A63