On lower bounds in radial basis approximation
In this paper, upper bounds on the obtainable approximation orders of radial basis function approximations to approximands from Sobolev spaces are established. For this, any radial basis function of the thin-plate spline type is considered as in earlier papers, especially by Johnson, but the results are improved. For instance, for certain choices of the Sobolev spaces (namely, \(W^k_\infty\)) and the uniform norm for the error estimate, exact orders for the approximation are shown by combining the new lower bounds on the error (i.e., upper bounds on the order of the error) with the known upper bounds on the error (i.e., lower bounds on its order). The centres by which the radial basis functions are shifted when these approximations are applied, are allowed to be scattered.
scientific article; zbMATH DE number 2140536
scientific article