Generalized Hermite Interpolation via Matrix-valued Conditionally Positive Definite Functions

In this paper, we consider a broad class of interpolation problems, for both scalarand vector-valued multivariate functions subject to linear side conditions, such as being divergence-free, where the data are generated via integration against compactly supported distributions. We show that, by using certain families of matrix-valued conditionally positive definite functions, such interpolation problems are well poised; that is, the interpolation matrices are invertible. As a sample result, we show that a divergence-free vector field can be interpolated by a linear combination of convolutions of the data-generating distributions with a divergence-free, 3x3 matrix-valued conditionally positive definite function. In addition, we obtain norm estimates for inverses of interpolation matrices that arise in a class of multivariate Hermite interpolation problems. 0. Introduction Background. The last few years have seen an increase in our theoretical understanding of methods for dealing with the problem of interpolating function values when the data sites are both scattered and multidimensional. Duchon's work on the method of thin-plate splines [5, 6] and the work of Micchelli [18] and Madych and Nelson [15, 16] on radial basis functions (RBFs) established the invertibility of the interpolation matrices associated with these functions; that is, their work established that the scattered-data interpolation problem was well poised—i.e., a solution to the problem exists and is unique—relative to the families of either the thin-plate splines or certain RBFs, such as the Hardy multiquadrics [10, 11] or the Gaussians. The Hardy multiquadrics have been applied extensively in surface fitting problems arising in geodesy, geophysics, and other areas [11], while the Gaussians have been employed in problems arising in connection with neural networks [1, 21, 22] and adaptive control [24]. A method for dealing with a Hermite interpolation problem—the problem of interpolation of data involving both function values and values of combinations of derivatives—when the data come from scattered, multidimensional sites was introduced by Hardy [11], who mentioned that he had had computational success with it. This method also arises at least implicitly in connection with the Received by the editor October 27, 1992 and, in revised form, September 14, 1993. 1991 Mathematics Subject Classification. Primary 41A05, 41A63, 41A29.