Scattered Data Interpolation on Spheres: Error Estimates and Locally Supported Basis Functions

Abstract

Error estimates for scattered data interpolation by "shifts" of a positive definite function for target functions in the associated reproducing kernel Hilbert space (RKHS) have been known for a long time. However, apart from special cases where data is gridded, these estimates do not apply when the target functions generating the data are outside of the associated RKHS, and in fact no estimates were known for such target functions. In this paper, working with the n-sphere as the underlying manifold, we obtain Sobolev-type error estimates for interpolating functions $f\in C^{2k}(S^n)$ from "shifts" of a smoother positive definite function $\phi$ defined on Sn. Moreover, the estimates are close to the optimal approximation order. We also introduce a class of locally supported positive definite functions on Sn, functions based on Wendland's compactly supported radial basis functions (RBFs) [H. Wendland, Adv. Comput. Math., 4 (1995), pp. 389--396], which can be both explicitly and easily computed and also analyzed for convergence properties.