Norm estimates for the inverses of a general class of scattered-data radial-function interpolation matrices

AbstractIn this paper we investigate those radial basis functions h associated with functions whose mth derivative (modulo a scalar multiple) is completely monotonic. Our results apply both to interpolation problems that require polynomial reproduction and to those that do not. In the case where polynomial reproduction is not required and the order m is 0 or 1, we obtain estimates on the norms of inverses of scattered-data interpolation matrices. These estimates depend only on the minimal-separation distance for the data and on the dimension of the ambient space, Rs. When the order m satisfies m ⩾ 2, we show that there exist parameters a1, …, am such that the function h(x) + am + am − 1r2 + … + a1r2m − 2 gives rise to an invertible interpolation matrix, and we obtain bounds on the norm of the inverse of this matrix. For interpolation methods in which one wishes to reproduce polynomials of total degree m − 1 or less, bounds for the norm of the inverse of the interpolation matrix are obtained, provided the data contains a πm-1(Rs) unisolvent subset. These results apply, in particular, to Duchon's “thin-plate spline” results